U.S. patent application number 10/771739 was filed with the patent office on 2004-10-21 for apparatus and methods for performing process simulation using a hybrid model.
This patent application is currently assigned to Moldflow Ireland Ltd.. Invention is credited to Kennedy, Peter, Yu, Huagang.
Application Number | 20040210429 10/771739 |
Document ID | / |
Family ID | 32869318 |
Filed Date | 2004-10-21 |
United States Patent
Application |
20040210429 |
Kind Code |
A1 |
Yu, Huagang ; et
al. |
October 21, 2004 |
Apparatus and methods for performing process simulation using a
hybrid model
Abstract
The invention provides an apparatus and methods for performing
process simulation and structural analysis using a hybrid model.
For example, a method of the invention automatically defines a
hybrid solution domain by dividing a representation of a plastic
component or mold cavity into two portions--a portion in which a
simplified analysis may be conducted, and a portion in which a more
complex analysis is required. The method may use as input any form
of CAD data that describes the surface of a component or mold.
Furthermore, the invention provides methods for simulating fluid
flow within a mold cavity by automatically creating a hybrid
solution domain, automatically discretizing the domain, and solving
for the distribution of process variables within the solution
domain.
Inventors: |
Yu, Huagang; (Bulleen,
AU) ; Kennedy, Peter; (Ithaca, NY) |
Correspondence
Address: |
TESTA, HURWITZ & THIBEAULT, LLP
HIGH STREET TOWER
125 HIGH STREET
BOSTON
MA
02110
US
|
Assignee: |
Moldflow Ireland Ltd.
Dublin
IE
|
Family ID: |
32869318 |
Appl. No.: |
10/771739 |
Filed: |
February 4, 2004 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
60445182 |
Feb 5, 2003 |
|
|
|
Current U.S.
Class: |
703/9 |
Current CPC
Class: |
G06F 2113/22 20200101;
B29C 45/7693 20130101; G06F 30/23 20200101; G06G 7/50 20130101 |
Class at
Publication: |
703/009 |
International
Class: |
G06G 007/48 |
Claims
What is claimed is:
1. A method for simulating fluid flow within a mold cavity, the
method comprising the steps of: (a) providing a surface
representation for a three-dimensional volume associated with a
mold cavity; (b) separating the surface representation into at
least a first portion and a second portion, the first portion of
the surface representation being associated with at least one
section of the volume having at least one of (i) a substantially
invariant thickness and (ii) a gradually varying thickness along a
length thereof; (c) discretizing a first portion of a solution
domain bound on an exterior thereof by the first portion of the
surface representation; (d) discretizing a second portion of the
solution domain bound on an exterior thereof by the second portion
of the surface representation; (e) defining a plurality of
interface elements for the solution domain that connect at least
part of the first portion of the solution domain to at least part
of the second portion of the solution domain; (f) obtaining values
of at least one process variable for the first portion of the
solution domain using a first set of governing equations; and (g)
obtaining values of the at least one process variable for the
second portion of the solution domain using a second set of
governing equations.
2. The method according to claim 1, wherein step (b) is performed
automatically.
3. The method according to claim 2, wherein at least one of step
(c), step (d), and step (e) is performed automatically.
4. The method according to claim 2, wherein at least two of step
(c), step (d), and step (e) are performed automatically.
5. The method according to claim 1, wherein the surface
representation is a surface mesh.
6. The method according to claim 1, wherein the volume represents a
molded object.
7. The method according to claim 1, wherein the volume represents a
mold cavity.
8. The method according to claim 1, wherein the first set of
governing equations in step (f), the second set of governing
equations in step (g), and a set of interface element equations are
solved simultaneously, subject to initial conditions and boundary
conditions.
9. The method of claim 8, wherein the interface element equations
link a portion of the solution domain described by governing
equations in two spatial dimensions to a portion of the solution
domain described by governing equations in three spatial
dimensions.
10. The method according to claim 1, wherein the at least one
process variable is selected from the group consisting of
temperature, pressure, fluid velocity, stress, and fluid flow front
position.
11. The method according to claim 1, wherein there are at least two
process variables selected from the group consisting of
temperature, pressure, fluid velocity, stress, and fluid flow front
position.
12. The method according to claim 1, wherein there are at least
three process variables including temperature, pressure, and fluid
velocity.
13. The method according to claim 1, wherein the method simulates
fluid injection in the three-dimensional volume.
14. The method according to claim 13, wherein the method further
comprises determining a location of at least one injection
point.
15. The method according to claim 1, wherein step (a) comprises
providing the surface representation from CAD system output.
16. The method according to claim 15, wherein the CAD system output
is in stereolithography format or IGES format.
17. The method according to claim 15, wherein the CAD system output
defines a surface mesh comprising polygonal elements.
18. The method according to claim 17, wherein the polygonal
elements are triangular elements or quadrilateral elements.
19. The method according to claim 15, wherein the CAD system output
defines a three-dimensional mesh.
20. The method according to claim 19, wherein the surface
representation is provided from a lattice of polygons that bound
the three-dimensional mesh.
21. The method according to claim 15, wherein step (a) comprises
using the CAD system output as a preliminary mesh and remeshing the
preliminary mesh to provide the surface representation.
22. The method according to claim 1, wherein step (a) comprises
providing a surface representation comprising a mesh of polygonal
surface elements.
23. The method according to claim 22, wherein step (b) comprises
defining two or more subsurfaces, each subsurface comprising at
least one of the surface elements.
24. The method according to claim 23, wherein step (b) comprises
determining element properties and nodal properties for each of the
surface elements.
25. The method according to claim 24, wherein step (b) comprises
using at least a subset of the element properties and nodal
properties to classify each of the two or more subsurfaces
according to curvature.
26. The method according to claim 23, wherein step (b) comprises
defining at least one surface loop, each comprising a connected
subset of edges of the surface representation.
27. The method according to claim 23, wherein step (b) comprises
remeshing at least a subset of the two or more subsurfaces using a
bisection algorithm.
28. The method according to claim 23, wherein step (b) comprises
determining which of the two or more subsurfaces are matched
subsurfaces.
29. The method according to claim 28, wherein each pair of matched
subsurfaces is separated by a definable thickness.
30. The method according to claim 28, wherein the first portion of
the surface representation comprises at least a subset of the
matched subsurfaces.
31. The method according to claim 23, wherein step (b) comprises
determining which of the two or more subsurfaces are unmatched
subsurfaces.
32. The method according to claim 31, wherein the second portion of
the surface representation comprises at least a subset of the
unmatched subsurfaces.
33. The method according to claim 23, wherein step (b) comprises
determining which of the two or more subsurfaces are edge
subsurfaces.
34. The method according to claim 28, wherein step (c) comprises
projecting at least one of the surface elements from one subsurface
in a substantially perpendicular direction onto a matched
subsurface thereof, thereby defining paired surface elements.
35. The method according to claim 34, wherein step (c) comprises
converting the paired surface elements into wedge elements.
36. The method according to claim 1, wherein step (c) comprises
automatically discretizing the first portion of the solution
domain.
37. The method according to claim 1, wherein discretizing in step
(c) comprises using the first portion of the surface representation
to define the first portion of the solution domain.
38. The method according to claim 1, wherein step (c) comprises
discretizing the first portion of the solution domain using wedge
elements.
39. The method according to claim 38, wherein at least one of the
wedge elements comprises at least one solution grid point along a
thickness thereof.
40. The method according to claim 38, wherein at least one of the
wedge elements is a discretely layered element or a continuously
layered element.
41. The method according to claim 1, wherein step (d) comprises
automatically discretizing the second portion of the solution
domain.
42. The method according to claim 1, wherein step (c) comprises
discretizing the first portion of the solution domain using
hexahedral elements.
43. The method according to claim 1, wherein step (c) comprises
discretizing the first portion of the solution domain using shell
elements.
44. The method according to claim 1, wherein step (d) comprises
discretizing the second portion of the solution domain using
polyhedral elements.
45. The method according to claim 44, wherein the polyhedral
elements are tetrahedral elements or hexahedral elements.
46. The method according to claim 1, wherein step (e) comprises
defining a set of line interface elements.
47. The method according to claim 46, wherein each of the line
interface elements is located along an interface of the first
portion of the solution domain and the second portion of the
solution domain.
48. The method according to claim 46, wherein each of the line
interface elements comprises at least two nodes of a wedge element
of the first portion of the solution domain.
49. The method according to claim 48, wherein each of the line
interface elements further comprises at least one solution grid
point between two of the at least two nodes.
50. The method according to claim 1, wherein step (e) comprises
defining a set of planar interface elements.
51. The method according to claim 1, wherein step (c) is initiated
before step (e).
52. The method according to claim 1, wherein step (e) is initiated
before step (d).
53. The method according to claim 1, wherein the first set of
governing equations describes fluid flow in two spatial
dimensions.
54. The method according to claim 1, wherein the first set of
governing equations describes fluid flow in two spatial dimensions
and time.
55. The method according to claim 1, wherein the first set of
governing equations describes fluid flow in one spatial dimension
and time.
56. The method according to claim 1, wherein step (f) comprises
using a Hele-Shaw approximation.
57. The method according to claim 1, wherein step (g) comprises
solving a Navier Stokes equation.
58. The method according to claim 1, wherein step (g) comprises
solving a simplified Stokes equation.
59. The method according to claim 1, wherein the second set of
governing equations comprises conservation of mass, conservation of
momentum, and conservation of energy equations.
60. The method according to claim 1, wherein at least one of step
(f) and step (g) comprises using a meshless scheme.
61. The method according to claim 60, wherein the meshless scheme
is a boundary element method, natural element method, or smooth
particle hydrodynamics method.
62. The method according to claim 1, further comprising the step
of: (h) displaying the values of the at least one process variable
directly on a 3D representation of the volume.
63. The method according to claim 1, wherein step (g) comprises
using a Mini element formulation.
64. A method for simulating fluid flow within a mold cavity, the
method comprising the steps of: (a) providing a surface
representation for a three-dimensional volume associated with a
mold cavity; (b) automatically separating the surface
representation into at least a first portion and a second portion;
(c) defining a solution domain for the three-dimensional volume,
where the solution domain comprises a first part corresponding to
the first portion of the surface representation and a second part
corresponding to the second portion of the surface representation;
(d) solving for a process variable in the first part of the
solution domain; and (e) solving for the process variable in the
second part of the solution domain.
65. The method according to claim 64, wherein the first portion of
the surface representation in step (b) is associated with at least
one section of the volume that has at least one of (i) a
substantially invariant thickness and (ii) a gradually varying
thickness along a length thereof.
66. The method according to claim 64, wherein step (c) comprises
automatically discretizing the first part and the second part of
the solution domain.
67. The method according to claim 64, further comprising the step
of defining a plurality of interface elements that connect the
first part of the solution domain to the second part of the
solution domain.
68. The method according to claim 64, wherein step (d) comprises
using a first set of governing equations and step (e) comprises
using a second set of governing equations.
69. The method according to claim 68, wherein the first set of
governing equations describes 2.5D flow and the second set of
governing equations describes 3D flow.
70. A method for automatically defining a hybrid solution domain,
the method comprising the steps of: (a) identifying a plurality of
subsurfaces of a volume associated with a mold cavity using a
representation of the surface of the volume; (b) matching one or
more pairs of the plurality of subsurfaces to identify one or more
matched pairs of subsurfaces and one or more unmatched subsurfaces;
and (c) defining (i) a first portion of a hybrid solution domain
bound at least in part by one or more of the matched pairs of
subsurfaces and (ii) a second portion of the hybrid solution domain
bound at least in part by one or more of the unmatched
subsurfaces.
71. The method according to claim 70, wherein the volume represents
a mold cavity.
72. The method according to claim 71, further comprising using the
hybrid solution domain to model a molding process.
73. The method according to claim 70, wherein the representation of
the surface of the volume comprises CAD system output.
74. The method according to claim 70, wherein the first portion of
the hybrid solution domain is amenable to 2.5D flow analysis, and
the second portion of the hybrid solution domain is amendable to 3D
flow analysis.
75. The method according to claim 70, wherein step (b) comprises
classifying each of the plurality of subsurfaces according to
curvature.
76. The method according to claim 70, wherein the matched pairs of
subsurfaces each comprise two subsurfaces that are separated by a
substantially constant thickness.
77. The method according to claim 70, wherein the volume represents
a molded object.
78. The method according to claim 77, further comprising using the
hybrid solution domain in determining a structural property of the
molded object.
79. The method according to claim 78, wherein the structural
property is warpage.
80. An apparatus for simulating fluid flow within a mold cavity,
the apparatus comprising: (a) a memory that stores code defining a
set of instructions; and (b) a processor that executes said
instructions thereby to (i) separate a surface representation of a
three-dimensional volume associated with a mold cavity into at
least a first portion and a second portion, the first portion of
the surface representation being associated with at least one
section of the volume having at least one of (i) a substantially
invariant thickness and (ii) a gradually varying thickness along a
length thereof; (ii) discretize a first portion of a solution
domain bound on an exterior thereof by the first portion of the
surface representation; (iii) discretize a second portion of the
solution domain bound on an exterior thereof by the second portion
of the surface representation; (iv) define a plurality of interface
elements for the solution domain that connect at least part of the
first portion of the solution domain to at least part of the second
portion of the solution domain; (v) obtain values of at least one
process variable for the first portion of the solution domain using
a first set of governing equations; and (vi) obtain values of the
at least one process variable for the second portion of the
solution domain using a second set of governing equations.
81. An apparatus for defining a hybrid solution domain, the
apparatus comprising: (a) a memory that stores code defining a set
of instructions; and (b) a processor that executes said
instructions thereby to (i) identify a plurality of subsurfaces of
a volume associated with a mold cavity using a representation of
the surface of the volume; (ii) match one or more pairs of the
plurality of subsurfaces to identify one or more matched pairs of
subsurfaces and one or more unmatched subsurfaces; and (iii) define
(A) a first portion of a hybrid solution domain bound at least in
part by one or more of the matched pairs of subsurfaces and (B) a
second portion of the hybrid solution domain bound at least in part
by one or more of the unmatched subsurfaces.
82. The method of claim 2, further comprising the step of
re-characterizing a subset of the second portion of the solution
domain as belonging to the first portion according to user
input.
83. The method of claim 2, further comprising the step of
re-characterizing a subset of the first portion of the solution
domain as belonging to the second portion according to user
input.
84. The method of claim 1, wherein step (b) comprises separating
the surface representation into a first portion, a second portion,
and at least one additional portion.
Description
PRIOR APPLICATIONS
[0001] The present application claims the benefit of U.S.
Provisional Patent Application No. 60/445,182, filed Feb. 5, 2003,
which is hereby incorporated by reference in its entirety.
FIELD OF THE INVENTION
[0002] This invention relates generally to methods of process
simulation and analysis. More particularly, the invention relates
to the simulation of injection molding using a multidimensional
model.
BACKGROUND OF THE INVENTION
[0003] Manufacturers use process analysis and structural analysis
in designing a wide variety of products, including consumer goods,
automotive parts, electronic equipment, and medical equipment. It
is often advantageous to simulate or otherwise model a
manufacturing process to aid in the development of a particular
product. A computer simulation of a manufacturing process may allow
accurate prediction of how changes in process variables and/or
product configuration will affect production. By performing process
simulation, a designer can significantly reduce the time and cost
involved in developing a product, since computer modeling reduces
the need for experimental trial and error. Computer-aided process
simulation allows for optimization of process parameters and
product configuration during the early design phase, when changes
can be implemented more quickly and less expensively.
[0004] A manufacturer may also use modeling to predict structural
qualities of a manufactured product, such as how the product will
react to internal and external forces after it is made. A
structural model may be used, for instance, to predict how residual
stress in a molded product may result in product warpage.
Structural models aid in the design of a product, since many
prospective versions of the design can be tested before actual
implementation. Time-consuming trial and error associated with
producing and testing actual prototypes can be greatly reduced.
[0005] There is increasing demand for uniquely designed components.
This is particularly true in the field of plastics manufacturing,
where uniquely adaptable materials may be formed into a myriad of
configurations using processes such as injection molding,
compression molding, thermoforming, extrusion, pultrusion, and the
like. This is also true in the manufacturing of parts made with
fiber-filled materials, composites, and other specialty materials,
custom-designed for specialized uses.
[0006] Process and structural analysis in these fields poses
significant challenges. For example, there is increasing demand for
products having complex geometries. In order to properly model a
molding process for a product having a complex geometry, the mold
must be adequately characterized by the solution domain of the
model. Modeling processes involving components with complex
geometries requires significantly more computational time and
computer resources than modeling processes involving components
with simple geometries.
[0007] Also, injection-molded plastic is viscoelastic and may have
highly temperature-dependent and shear-dependent properties. These
complexities further increase computational difficulty of process
and structural simulations involving plastic components. Governing
equations of adequate generality must be solved over complex
domains, taking into account the changing properties of the
material being processed. Analytical solutions of these equations
over complex domains are generally unavailable; thus, numerical
solutions must be sought.
[0008] Computer models use numerical methods to approximate the
exact solution of governing equations over complex geometries,
where analytical solutions are unavailable. A model of an injection
molding process may include, for example, a solution domain in the
shape of the mold interior, discretized to enable accurate
numerical approximation of the solution of the applicable governing
equations over the solution domain.
[0009] Process models often simulate molds having complicated
shapes by using solution domains with simplified geometries,
thereby reducing required computation time and computer resources.
For example, certain injection molding process simulators use a
two-dimensional (2D) solution domain to simplify the geometry of
the real, three-dimensional (3D) mold, thereby greatly reducing
computational complexity. Many of these simulators use a Hele-Shaw
solution approach, where pressure variation and fluid flow in the
thickness direction are assumed to be zero. These "2.5D" models are
generally beneficial for simulating injection molding of
thin-walled components having relatively simple geometries.
However, in components that have thick portions or complex
geometries, injected material flows in all three directions, and
traditional thin-wall assumptions do not apply, making the 2.5D
analysis inadequate.
[0010] Current 3D models of injection molding processes do not make
thin-wall assumptions; they solve constitutive equations over a
three-dimensional solution domain. These models are computationally
complex, generally requiring significantly greater computer
resources and computation times for process simulation than the
simpler 2.5D models. Three-dimensional models of injection molding
processes generally use a finite element scheme in which the
geometry of the mold is simulated with a mesh of 3D elements. The
size of the elements, or the discretization, required to accurately
model a given process depends on the geometry of the solution
domain and the process conditions. The generation of a 3D mesh is
not trivial, and there is currently no consistent method of
automatically generating a suitable 3D mesh for a given
application.
[0011] Determining a suitable mesh for a 2.5D, Hele-Shaw-based
model is also non-trivial. For example, it is typically necessary
to define a surface representing the midplane of a thin-walled
component, which is then meshed with triangular or quadrilateral
elements to which appropriate thicknesses are ascribed. Thus, there
is an added step of determining a midplane surface that must be
performed after defining solution domain geometry.
[0012] Many manufactured components have at least some portion that
is thin-walled or shell-like, that may be amenable to simulation
using a 2.5D model. However, many of these components also have one
or more thick or complex portions in which the 2.5D assumptions do
not hold, thereby making the overall analysis inaccurate. One may
use a 3D model to more comprehensively simulate processing of
components that have both thick and thin portions. However, the
computational complexity of a 3D model is much greater than that of
a 2.5D model, thereby increasing the time and computer resources
required for analysis.
[0013] Additionally, the way a 3D model must be discretized further
reduces the efficiency of a 3D process model for a component having
thin portions. For example, a typical thin portion of a molded
component may have a thickness of about 2 mm, whereas the length of
the thin portion may be hundreds of millimeters. During the molding
process, there will generally be a large thermal gradient across
the thickness of the thin portion, perhaps hundreds of degrees per
millimeter, whereas the temperature gradient along the length of
the portion (transverse to the thickness) may be extremely low.
Conversely, the pressure gradient in the thickness direction will
generally be very low, while the pressure gradient in the
transverse direction will be very high. The high variability of
these properties in at least two directions--temperature across the
thickness, and pressure along the length--calls for a very dense
mesh with many solution nodes in order to achieve an accurate
process simulation, thereby increasing computational complexity.
Thus, the time required for accurate 3D simulation of a typical
component containing both a thick and a thin portion may be as much
as a day or more and may require significant computer resources,
due to the fine discretization required.
[0014] Hybrid simulations solve simplified flow equations in the
relatively thin regions of a given component and more complex flow
equations in other regions. Hybrid simulations may reduce the
computational complexity associated with full 3D models while
improving the simulation accuracy associated with 2.5D models.
[0015] A hybrid solution scheme has been proposed in Yu et al., "A
Hybrid 3D/2D Finite Element Technique for Polymer Processing
Operations," Polymer Engineering and Science, Vol. 39, No. 1, 1999.
The suggested technique does not account for temperature variation
and, thus, does not provide accurate results in non-isothermal
systems where material properties vary with temperature, as in most
injection molding systems. Example applications of the technique
involve relatively simple solution domains that have been
pre-divided into "2D" and "3D" portions. Furthermore, there does
not appear to be a suggestion of how to adapt the technique for the
analysis of more complex parts than the examples shown.
[0016] U.S. Pat. No. 6,161,057, issued to Nakano, suggests a simple
hybrid solution scheme that solves for process variables in a thick
portion and a thin portion of a solution domain. The suggested
technique requires simplifying assumptions to calculate pressure
and fluid velocity in both the thick and thin portions of the
solution domain. For example, the technique requires using Equation
1, below, to calculate fluid velocity in the thick portion of the
solution domain: 1 x = P x , y = P y , z = P z ( 1 )
[0017] where .upsilon..sub.x, .upsilon..sub.y, and .upsilon..sub.z
are fluid velocity in the x, y, and z directions, respectively; P
is pressure; and .xi. is flow conductance, which is defined in the
Nakano patent as a function of fluid viscosity. The approximation
of Equation 1 is more akin to the 2.5D Hele Shaw approximation than
full 3D analysis, and Equation 1 does not adequately describe fluid
flow in components having thick and/or complex portions,
particularly where the thick portion makes up a substantial
(nontrivial) part of the component.
[0018] Current modeling methods are not robust; they must be
adapted for use in different applications depending on the
computational complexity involved. Modelers decide which modeling
method to use based on the process to be modeled and the geometry
of the component to be produced and/or analyzed. Modelers must also
determine how to decompose a solution domain into elements
depending on the particular component and process being simulated.
The decisions made in the process of choosing and developing a
model for a given component and/or process may well affect the
accuracy of the model output. The process of adapting models to
various applications is time-consuming and generally involves
significant customization by a highly-skilled technician.
[0019] There is a need for a more accurate, more robust, faster,
and less costly method of modeling manufacturing processes and
performing structural analyses of manufactured components. Current
methods require considerable input by a skilled technician and must
be customized for the component and/or process being modeled.
SUMMARY OF THE INVENTION
[0020] The invention provides an apparatus and methods for using
CAD system data to automatically define a hybrid analysis solution
domain for a mold cavity and/or molded component. The invention
also provides an apparatus and methods for simulating the molding
of a manufactured component using a hybrid analysis technique.
[0021] The invention overcomes the problems inherent in current
hybrid analysis systems, which require intervention by a skilled
technician to define a solution domain from CAD system output. The
invention provides an automatic, standardized method of defining a
hybrid solution domain from CAD system output without requiring
expert human intervention. The invention also provides hybrid
process analysis techniques that offer improvements upon prior
techniques, for example, by accounting for temperature variation
and/or complex flow behaviors.
[0022] Simulation of fluid flow within a mold cavity generally
requires a representation of the mold cavity or molded component.
In one aspect, the invention provides a method for simulating fluid
flow that automatically divides a representation of a component
and/or mold cavity into at least two portions--a portion in which a
simplified analysis may be conducted, and a portion in which a more
complex analysis is required. The method then includes performing a
hybrid analysis--that is, solving a set of simplified governing
equations in the simpler portion and a set of more complex
governing equations in the complex portion. This reduces the amount
of time and memory required to perform a simulation, without
compromising accuracy, since the complex set of equations must be
solved only where the geometry of the mold or component is
complicated. The simplified analysis may be a 2.5D Hele-Shaw
analysis, a 2D analysis, a 1D analysis, or any other kind of
analysis in which simplifying assumptions can be made with respect
to one or more dimensions and/or other variables.
[0023] In one embodiment, the invention automatically separates a
representation of the geometry of a manufactured component or mold
into at least two portions--a portion for 2.5D analysis and a
portion for 3D analysis. For example, the invention may use a
surface representation of a manufactured component or mold to
define a solution domain for hybrid analysis, where the domain is
automatically separated into one or more 2.5D-analysis portions and
one or more 3D-analysis portions. The 2.5D-analysis portions of the
domain each have a substantially invariant or gradually-varying
thickness, while the 3D-analysis portions generally have a more
complex geometry. For example, the method may identify thin-walled
portions of a manufactured component for 2.5D analysis, and
separate these from more complex portions--such as corners, the
bases of ribs, and intersections of surfaces--for which 3D analysis
is performed. In one embodiment, the method also automatically
discretizes the 2.5D-analysis portions and the 3D-analysis portions
of the solution domain and solves for the distribution of process
variables--such as pressure, velocity, and temperature--as
functions of time.
[0024] The invention provides simulations having greater accuracy
than current hybrid schemes. For example, an embodiment of the
invention accounts for temperature by incorporating an energy
conservation equation in the analysis. Furthermore, the invention
allows solution of accurate forms of the mass and momentum
conservation equations in the analysis scheme, without requiring
simplifying assumptions, as in Equation 1.
[0025] Methods of the invention provide faster, less costly
simulations than traditional 3D solution techniques, since a full
3D analysis is only performed where necessary. For example, in one
embodiment, the invention analyzes as much of the domain as
possible--for example, thin, flat portions of the domain--with a
simpler, 2.5D scheme, with negligible impact on accuracy.
[0026] Methods of the invention are more robust and require less
input from skilled technicians than traditional simulation
techniques. For example, in one embodiment, the invention uses
simple CAD system output to define a surface mesh of a component or
mold to be modeled, then automatically divides the mesh into a
2.5D-analysis portion and a 3D-analysis portion via a subsurface
matching technique, and automatically discretizes the two portions
to form a solution domain in which hybrid analysis is performed. It
is not necessary for a technician to decide how to separate a
solution domain into 2.5D and 3D analysis portions, since the
embodiment performs the separation automatically. In addition to
CAD system output, the invention may use any other type of data
file conveying a representation of the surface of the component or
mold to be modeled. Since the domain is tied to the actual geometry
of the component or mold surface, the invention is capable of
displaying results directly on the 3D geometry of the component,
making interpretation of results more intuitive for a user than
schemes which require the creation of a midplane mesh, for
example.
[0027] In some cases it is useful to allow a user to exert control
over the automatically-decomposed solution domain. For example, the
automatic decomposition of a given surface domain into a hybrid
solution domain may result in regions that are classified as part
of the complex portion (i.e. 3D-analysis portion), in which it may
be reasonable to perform a simpler analysis (i.e. 2.5-D analysis).
For example, a user may wish to tolerate some reduction in accuracy
in order to increase analysis speed during the early stages of
design, where more accurate analysis may be performed later. In
another example, a user may wish to increase simulation accuracy at
the expense of the computer time required. Therefore, one
embodiment of the invention allows a user to manually
re-characterize a given region that has been automatically
characterized as falling within either the first portion or the
second portion of the solution domain.
[0028] The method may also or alternatively allow a user to
manually characterize part of the volume to be analyzed as either
belonging to the first portion or the second portion of the
solution domain prior to the automatic decomposition. This may be
useful where the user knows that she/he would like a particular
kind of analysis (2.5D, 3D, etc.) in a given region of the
volume.
[0029] Although descriptions of certain embodiments of the
invention include the decomposition of a solution domain into a
first and a second portion, it is within the scope of the invention
to further decompose the solution domain into a third, fourth,
fifth, or additional portions in which different types of analysis
are to be performed.
[0030] Thus, in one aspect, the invention defines a surface
representation from user-provided CAD output; separates the surface
representation into two or more portions by analyzing and matching
subsurfaces; discretizes the two or more portions; and solves for
the distribution of one or more process variables--such as
pressure, velocity, and temperature--as a function of time. The
process being modeled may be the filling phase and/or packing phase
of an injection molding process, for example. The two or more
portions may include one or more 2.5D-analysis portions and one or
more 3D-analysis portions. The 2.5D-analysis portions of the
solution domain may be discretized with wedge elements, and the
3D-analysis portions of the solution domain may be discretized with
tetrahedral elements. Dual domain elements of the type discussed in
U.S. Pat. No. 6,096,088, to Yu et al., the disclosure of which is
incorporated by reference herein in its entirety, may be used
instead of wedge elements in the 2.5D-analysis portion. Hexahedral
elements may be used instead of tetrahedral elements in the
3D-analysis portion. Other types of elements may be used instead of
or in addition to those above. Furthermore, either or both of the
2.5D analysis and the 3D analysis may be performed using a
technique other than a finite element technique, such as a boundary
element method (BEM), a natural element method (NEM), smooth
particle hydrodynamics (SPH), or other meshless scheme.
[0031] Interface elements provide a link between the
simplified-analysis portions and the complex-analysis portions of a
solution domain. In one embodiment, conservation equations and
continuity requirements are enforced at the boundary between
2.5D-analysis portions and 3D-analysis portions using interface
elements. The interface elements are co-linear sets of nodes or
surfaces at the boundaries between the two types of portions of the
solution domain. In one embodiment, the interface elements are line
elements. In the case of structural analysis, an embodiment of the
invention uses interface elements to satisfy continuity
requirements and/or to match degrees of freedom at interfaces
between the two portions of the solution domain.
[0032] The invention provides a method for simulating fluid flow
within a mold cavity that includes the steps of providing a surface
representation of a mold cavity or molded component; automatically
separating the surface representation into at least a first portion
and a second portion; defining a solution domain corresponding to
the first and second portions; and solving for one or more process
variables in both portions of the solution domain.
[0033] In one embodiment, one or more steps of the method are
performed automatically in the sense that they are performed by
computer, requiring limited or no input from a skilled technician.
For example, in one embodiment, a discretized, hybrid solution
domain is produced automatically from a user-provided description
of the surface of a component or mold cavity, without requiring
additional input from the user. In another example, a discretized,
hybrid solution domain is produced automatically from a
user-provided description of the surface of a component or mold
cavity, where the user also provides (or is prompted to provide)
information regarding element aspect ratio, specified edge length
(SEL), process model inputs such as boundary conditions and/or
initial conditions, and/or other information related to how the
solution domain will be used. The production of the solution domain
is still automatic, even though a user provides certain
specifications, since the separation of the domain into portions
and the discretization of the solution domain are subject to
internal constraints imposed by the computer-performed method.
Certain embodiments provide default values of one or more modeling
specifications for which the user is prompted. The default values
may or may not be based on the specific component and/or process
being modeled. Certain embodiments provide a user the option of
providing a modeling specification himself, accepting a
pre-determined default value of the modeling specification, and/or
using a computer-determined value of the modeling specification
based on information about the component and/or process being
modeled.
[0034] In another aspect, the invention provides a method for
automatically defining a hybrid solution domain that includes the
steps of dividing a surface representation of a mold cavity or
molded component into subsurfaces; matching pairs of subsurfaces,
where the two subsurfaces of a given pair are separated by a
substantially constant or gradually-varying thickness (but where
the separation thickness of one pair may differ from that of
another); and defining a hybrid solution domain having a first
portion bound at least partly by the matched subsurfaces and a
second portion bound at least partly by one or more of the
unmatched subsurfaces. In one embodiment, the first portion is
amenable to 2.5D analysis while the second portion requires 3D
analysis for accurate solution. The method may further comprise
using the hybrid solution domain to model a molding process such as
injection molding or to determine a structural property of a molded
object, such as the warpage of a molded plastic component.
[0035] The invention also provides an apparatus for simulating
fluid flow within a mold cavity, as well as an apparatus for
defining a hybrid solution domain. Each apparatus includes a memory
that stores code defining a set of instructions, and a processor
that executes the instructions to perform one or more methods of
the invention described herein.
BRIEF DESCRIPTION OF THE DRAWINGS
[0036] The objects and features of the invention can be better
understood with reference to the drawings described below, and the
claims. The drawings are not necessarily to scale, emphasis instead
generally being placed upon illustrating the principles of the
invention. In the drawings, like numerals are used to indicate like
parts throughout the various views. The patent or application file
contains at least one drawing executed in color. Copies of this
patent or patent application publication with color drawing(s) will
be provided by the U.S. Patent and Trademark Office upon request
and payment of the necessary fee.
[0037] FIG. 1 depicts the solution domain used to simulate
injection molding of a component, where the domain is discretized
using tetrahedral elements for full 3D analysis, according to an
illustrative embodiment of the invention.
[0038] FIG. 2 depicts a portion of the solution domain used to
simulate injection molding of the component in FIG. 1, where the
portion of the domain is discretized using wedges for 2.5D analysis
or application of the dual domain method, according to an
illustrative embodiment of the invention.
[0039] FIG. 3 depicts a portion of the solution domain used to
simulate injection molding of the component in FIG. 1, where the
portion of the domain is discretized using tetrahedral elements for
3D analysis, according to an illustrative embodiment of the
invention.
[0040] FIG. 4 depicts a hybrid mesh solution domain used to
simulate injection molding of the component in FIG. 1, where one
portion of the domain is discretized using wedges for 2.5D
analysis, and another portion of the domain is discretized using
tetrahedral elements for 3D analysis, according to an illustrative
embodiment of the invention.
[0041] FIG. 5 depicts the pressure distribution at the
filling/packing switchover point, obtained using the hybrid mesh
solution domain of FIG. 4 and a combined 2.5D/3D approach,
according to an illustrative embodiment of the invention.
[0042] FIG. 6 is a schematic flow diagram depicting components in a
system for automatically defining a hybrid mesh solution domain,
used to solve for the distribution of process variables as
functions of time according to an illustrative embodiment of the
invention.
[0043] FIG. 7A depicts two adjacent elements on two different
subsurfaces, as analyzed in a system for defining a hybrid mesh
according to an illustrative embodiment of the invention.
[0044] FIG. 7B depicts the determination of bending angle between
the two adjacent elements from FIG. 7A according to an illustrative
embodiment of the invention.
[0045] FIG. 8A depicts two adjacent elements on two different
subsurfaces, as analyzed in a system for defining a hybrid mesh
according to an illustrative embodiment of the invention.
[0046] FIG. 8B depicts the determination of curvatures associated
with the two adjacent elements from FIG. 8A according to an
illustrative embodiment of the invention.
[0047] FIG. 9 depicts the application of criteria to limit
remeshing of subsurfaces in a system for defining a hybrid mesh
according to an illustrative embodiment of the invention.
[0048] FIG. 10 shows the discretization of a subsurface at
sequential stages of a remeshing procedure according to an
illustrative embodiment of the invention.
[0049] FIG. 11 shows an initial stereolithography surface
representation used in a remeshing procedure according to an
illustrative embodiment of the invention.
[0050] FIG. 12 shows the surface representation of FIG. 11
following remeshing according to an illustrative embodiment of the
invention.
[0051] FIG. 13 depicts a cross-section of a three-dimensional
T-shaped object and illustrates matching individual subsurfaces of
the surface representation of the object to categorize the
subsurfaces as matched, unmatched, or edge subsurfaces, according
to an illustrative embodiment of the invention.
[0052] FIG. 14 depicts a cross-section of a tapered, T-shaped
object and illustrates matching individual subsurfaces of the
surface representation of the object to categorize the subsurfaces
as matched, unmatched, or edge subsurfaces, according to an
illustrative embodiment of the invention.
[0053] FIG. 15 depicts criteria used in matching subsurfaces in a
system for defining a hybrid mesh according to an illustrative
embodiment of the invention.
[0054] FIGS. 16A, 16B, and 16C depict steps in a collapsing
procedure for categorizing subsurfaces as matched, unmatched, and
edge subsurfaces, according to an illustrative embodiment of the
invention.
[0055] FIG. 17 depicts a hybrid mesh solution domain comprising two
portions separated by interface elements, used in simulating fluid
flow within a mold cavity according to an illustrative embodiment
of the invention.
[0056] FIG. 18 depicts a portion of the hybrid mesh solution domain
of FIG. 17 comprising tetrahedral elements and interface elements,
used in simulating fluid flow within a mold cavity according to an
illustrative embodiment of the invention.
[0057] FIG. 19 depicts a plastic component of an automobile
dashboard; a hybrid solution domain is automatically determined and
an injection molding flow analysis is performed for the component
according to an illustrative embodiment of the invention.
[0058] FIG. 20 is a graphical representation of STL-formatted CAD
output produced during the design of the plastic component in FIG.
19, according to an illustrative embodiment of the invention.
[0059] FIG. 21 depicts a hybrid mesh solution domain used to
simulate injection molding of the component in FIG. 19, where one
portion of the domain is discretized using wedge elements for 2.5D
analysis or dual domain analysis, and another portion of the domain
is discretized using tetrahedral elements for 3D analysis,
according to an illustrative embodiment of the invention.
[0060] FIG. 22 depicts a map of times at which the flow front
reaches points within the mold of the component of FIG. 19,
obtained using the hybrid solution domain of FIG. 21 and a combined
2.5D/3D flow analysis approach, according to an illustrative
embodiment of the invention.
[0061] FIG. 23 depicts the pressure distribution at the
filling/packing switchover point, obtained using the hybrid mesh
solution domain of FIG. 21 and a combined 2.5D/3D flow analysis
approach, according to an illustrative embodiment of the
invention.
[0062] FIG. 24 depicts a computer hardware apparatus suitable for
use in carrying out the methods described herein, according to an
illustrative embodiment of the invention.
DETAILED DESCRIPTION
[0063] Table 1 lists various symbols used herein and is provided as
a convenience for the reader. Entries in Table 1 do not serve to
limit interpretation of embodiments of the invention described
herein.
1TABLE 1 Notation Symbol Description .gradient. Gradient operator
.alpha. 2 Thermal diffusivity : c P .beta. 3 Expansivity : 1 V V T
{dot over (.delta.)} Shear rate .eta. Viscosity .kappa. 4 C
ompressibility : 1 V V P .lambda. Thermal conductivity .nu.
Poisson's ratio .rho. Density .sigma. Stress .tau. Stress
(deviatoric part) .upsilon. Velocity {right arrow over (.upsilon.)}
Velocity vector .upsilon..sub.x, .upsilon..sub.y, .upsilon..sub.z
Velocity components .omega. Relaxation factor .xi. Flow
`conductance` .DELTA. Increment, i.e., in timestep .LAMBDA.
Locally-defined quantity .OMEGA. Domain .differential..OMEGA.
Boundary of domain, Mold/plastic interface c.sub.p Specific heat d
Distance of (tet) node from model wall e Element .differential.e
Boundary of element g, {right arrow over (g)} Gravity/body-force
and gravity/body-force vector h Half wall thickness of plastic
component/part k Coefficient matrix of element n Unit normal vector
r Load vector of element t Time A Area A.sub.e Area of element 5 A
e T , A e B Top and bottom area of a (shell/wedge) element F Frozen
fraction K Coefficient matrix of domain L Laplace coefficients for
element L.sub.ij 6 Component of Laplace coefficients L ij = N i x k
N j x k for element N Interpolation functions for element N.sub.i
Component of interpolation functions for element P Pressure Q Heat
flux R Load/residual vector R.sub.i Component of load/residual
vector S Flow conductance S.sub.e Elemental flow conductance T
Temperature T.sub.w Mold wall temperature V Volume V.sup.e Volume
of element 7 V n j e Nodal volume component of element W Weighting
factor Operator or Function Description i, j, k, l, m, n Integer
indices, i.e. node number, iteration, space dimension {right arrow
over (v )} Vector quantity, i.e. over space dimensions v.sub.i
Component of vector {overscore (K)} 1D matrix, i.e. over all nodes
8 K _ _ 2D matrix k.sub.ij Component of 2D matrix 9 K _ _ T
Transpose of matrix 10 k ij T Component of transpose of matrix
.gradient. Gradient operator 11 a b Dot product, i.e. .gradient.
.multidot. (.lambda..gradient.T) 12 a .times. b Cross product, i.e.
.gradient. .times. {right arrow over (S)} 13 a b Scalar product of
two tensors, c.sub.ijd.sub.ji 14 c :: d Tensor product of two
tensors, c.sub.ikd.sub.kj .delta..sub.ij Kronecker delta Element
equation quantity Description S.sub.i Nodal value of scalar field S
S.sup.e Element value of scalar field S 15 S i Nodal value of
vector field S S.sub.ij Nodal component (j = 1, 2, 3) of vector
field S 16 S e Element value of vector field S 17 S j e Element
component (j = 1, 2, 3) of vector field S .eta..sup.e Viscosity of
element (effective viscosity of frozen and molten portions in
parallel) 18 n e Unit normal surface of element e 19 r e Load
vector of element e (RHS) 20 V e Q Volume integral of scalar Q over
element e 21 Q , N e 22 V e QN 23 S e Q S Surface integral of
scalar Q over element e 24 W i e Weighting factor at node i of
element e
[0064] In general, the invention relates to an apparatus and
methods for performing process simulation and structural analysis
using a hybrid model. A hybrid model performs both 2.5D analysis
and 3D analysis in respective portions of a solution domain. Full
three-dimensional analysis of molded parts is often not possible
due to constraints on computer memory and CPU time. For example, in
order to mesh a plastic component or mold cavity for full 3D
analysis, it is often necessary to exceed the limit of addressable
memory that is available on a personal computer with a 32-bit
Windows operating system. The requirement for computer memory and
CPU time increases for the analysis of fiber-filled components
(parts), the analysis of injection molding, and the analysis of
warpage of injection molded parts after exposure at elevated
temperatures.
[0065] Thus, in one aspect, the invention provides an apparatus and
methods that automatically divide a representation of a component
or mold cavity into a 2.5D-analysis portion and a 3D-analysis
portion via a subsurface matching technique, and that automatically
discretize the two portions to form a solution domain in which
hybrid analysis is performed. Since many molded components contain
thin areas in which 2.5D analysis is appropriate, run times and
memory requirements are greatly decreased for many
applications.
[0066] FIG. 1 through FIG. 5 provide an introductory demonstration
that shows the simplification provided by application of an
embodiment of the invention to simulate an injection molding
process. FIG. 1 depicts the solution domain 100 for simulating
injection molding of a plastic component or mold cavity, determined
using a traditional technique. The solution domain 100 of FIG. 1 is
discretized using tetrahedral elements for full 3D analysis.
Overall discretization is complex, since the entire domain is made
up of three-dimensional tetrahedral elements, sized sufficiently
small so that simulation is accurate. For example, values of
process variables such as temperature, pressure, and fluid
velocity, may vary more in geometrically complex portions of the
domain, thus requiring smaller discretization for accurate
simulation in these regions. In addition to the dense
discretization, the full 3D governing equations must be solved
throughout the entire domain in order to solve for distribution of
process variables using the solution domain 100 of FIG. 1.
[0067] Certain portions of the solution domain 100 of FIG. 1 are
thin sections, each having a relatively constant thickness. It is
not necessary to perform full 3D analysis in these sections. For
example, methods of the invention can automatically divide a
surface representation of a component or mold cavity into a portion
having pieces each with a relatively constant thickness, where this
portion can be modeled using a 2.5D analysis technique. FIG. 2
depicts a portion of the solution domain 200 of the component/mold
cavity in FIG. 1 which can be modeled using a 2.5D analysis
technique. This 2.5D-analysis portion 200 includes pieces 202, 204,
206, 208, and 210, each having relatively constant thickness
(although the thickness of one piece may differ from the thickness
of another piece). The 2.5D-analysis portion is automatically
discretized using 6-node wedges. Alternatively, dual domain
elements can be used instead of wedges.
[0068] The remaining portion of the solution domain for the
component/mold cavity is automatically discretized using 3D
elements, such as tetrahedral elements, suitable for 3D analysis.
FIG. 3 depicts the portion 300 of the solution domain of the
component/mold cavity in FIG. 1 that is discretized using 3D
tetrahedral elements. This 3D-analysis portion 300 includes pieces
302, 304, 306, 308, 310, and 312. These pieces represent portions
of the component/mold cavity at the intersection of two or more
surfaces, at corners, and at other locations where thickness varies
suddenly.
[0069] FIG. 4 depicts a hybrid mesh solution domain 400 used to
simulate injection molding of the component/mold cavity in FIG. 1.
The solution domain 400 contains pieces of relatively constant
thickness, shown in FIG. 2, as well as the remaining pieces, shown
in FIG. 3. The hybrid solution scheme solves for the distribution
of one or more process variables--such as pressure, velocity, and
temperature--as a function of time, throughout the solution domain
400. The scheme includes use of a 2.5D analysis technique for the
wedge elements and a 3D analysis technique for the tetrahedral
elements. Interface elements lie at the boundary between the wedge
elements and the tetrahedral elements, where conservation equations
and continuity requirements are enforced.
[0070] FIG. 5 depicts pressure distribution 500 at a particular
point in time during the injection molding of the illustrative
component/mold cavity in FIG. 1 for which a hybrid solution domain
is determined and a combined 2.5D/3D flow analysis is performed
according to an embodiment of the invention. In this case, the
point in time corresponding to the pressure distribution 500 shown
is the switchover from the filling phase of injection molding to
the packing phase. The method for automatically determining the
hybrid solution domain and the method of performing the combined
2.5D/3D flow analysis is described in more detail herein below.
[0071] FIG. 6 is a schematic flow diagram 600 depicting components
in a system for automatically defining a hybrid mesh solution
domain, which is used to solve for the distribution of process
variables as functions of time. Embodiments of the invention also
include systems that automatically define a hybrid mesh solution
domain without necessarily using the solution domain to solve for
process variables. The system includes components for preprocessing
CAD system output 604 to provide an overall surface mesh; analyzing
surface elements 608 of the surface mesh; locating feature edges
612 of the surface mesh; classifying subsurfaces 616 of the surface
mesh; remeshing 620 the surface mesh; matching subsurfaces 624 of
the surface mesh, thereby identifying portions of the solution
domain in which 2.5D analysis can be performed; discretizing the
2.5D-analysis portion 628 of the solution domain; locating
interface elements 632; discretizing the 3D-analysis portion 636 of
the solution domain; and solving the governing equations 642
subject to initial conditions, boundary conditions, and process
inputs to obtain the solution for the distribution of process
variables throughout the solution domain, as functions of time.
Each of these components are discussed in more detail herein
below.
[0072] The preprocessor component 604 in FIG. 6 uses as input a
geometric description of a component to be manufactured, for
example, CAD output 602, and turns it into a representation for the
surface of the three-dimensional component/part or mold cavity 606.
Input can be any convenient form of geometric description. For
example, the preprocessor component 604 can use CAD system output
in Initial Graphics Exchange Specification (IGES) format (for
example, IGES Version 5.3, as well as later and earlier versions).
In another example, the component 604 uses a CAD system output file
associated with a common geometry kernel, such as Parasolids.RTM.
or ACIS.RTM.. Furthermore, the component 604 may use a CAD system
output file associated with a proprietary geometry kernel, such as
Pro-Engineer.RTM. from Parametric Technology Corporation of
Needham, Mass., or I-DEAS.RTM. from Structural Dynamics Research
Corporation of Milford, Ohio. A further means of usable CAD system
output includes stereolithography (STL) formatted files, used in
creating 3D prototypes. This format consists of planar triangles
with no connectivity in the finite element sense. Any file format
that describes a mesh covering the outer surfaces of a
three-dimensional solid region may be used as input. A remesher
(i.e. see component 620 in FIG. 6, discussed below) is used to
improve the mesh quality for subsequent processing.
[0073] The preprocessor 604 of FIG. 6 meshes the outer surfaces of
the three-dimensional part/component/mold cavity with a surface
mesh, for example, a mesh of triangular surface elements. Such a
mesh is frequently available from a CAD system using a geometry
kernel. For CAD output in stereolithography format, it is generally
necessary to remesh the part/component/mold cavity surface
representation to create a set of triangles with a reasonable
aspect ratio. A method of remeshing is depicted in component 620 of
FIG. 6, discussed in more detail herein below. Alternatively, the
surface representation provided by the preprocessor component 604
is made up of quadrilateral elements. Other two-dimensional
elements are also possible.
[0074] Once the preprocessor 604 in FIG. 6 produces a surface mesh
of triangular surface elements, a surface element analyzer 608
determines properties of the surface elements, and stores them for
later use. The surface element properties are used to divide the
surface mesh into subsurfaces, which are then classified and
matched to determine a first portion of the solution domain where
simplified analysis (i.e. 2.5D analysis) is sufficient. The surface
element analyzer 608 of FIG. 6 determines element properties and
nodal properties 610 associated with each of the surface elements.
For example, the surface element analyzer 608 computes and stores
the following element properties for each of the surface
elements:
[0075] Area;
[0076] Normal at the element centroid;
[0077] Edge lengths of the element;
[0078] Internal angles at the vertices of the element;
[0079] Adjacent elements (contiguous elements);
[0080] Bending angle between adjacent elements (the angle between
normals of adjacent elements); and
[0081] Bending curvature between adjacent elements (the curvature
of a cylindrical surface on which the bending edge and nodes of the
adjacent element lie).
[0082] FIGS. 7A and 7B demonstrate computation of bending angle
between adjacent elements, determined by the surface element
analyzer 608 of FIG. 6. Elements E1 and E2 of FIG. 7A happen to lie
along a feature edge 708 separating two subsurfaces 704, 706. FIG.
7B depicts a cross-sectional view of elements E1 and E2 as viewed
in the direction of arrow 702 in FIG. 7A. The bend angle between
adjacent elements E1 and E2 is shown in FIG. 7B as the angle
between the normals of elements E1 and E2.
[0083] FIGS. 8A and 8B demonstrate computation of bending curvature
(or "element curvature") between adjacent elements, determined in
the surface element analyzer 608 of FIG. 6. FIG. 8B depicts a
cross-sectional view of elements E1 and E2 as viewed in the
direction of arrow 702 in FIG. 8A. Bending (element) curvature
associated with element E1 on the edge E1 and E2 equals 1/R1, while
bending (element) curvature associated with element E2 on the edge
E1 and E2 equals 1/R2, as shown in the diagram 800 of FIG. 8B. The
bending curvature is the curvature of a cylindrical surface on
which the bending edge and nodes of the adjacent element can
lie.
[0084] In addition to element properties, the surface element
analyzer 608 of FIG. 6 determines nodal properties 610 associated
with each of the surface elements of the surface mesh 606. In one
embodiment, the following nodal properties are computed and stored
for each of the surface elements:
[0085] Measure of minimum curvature at the node;
[0086] Connecting elements (the set of elements connected to the
node); and
[0087] Number of edges connected to the node.
[0088] The minimum curvature at a given node is the smallest
bending curvature of all the elements attached to the given
node.
[0089] The feature edge locator 612 of FIG. 6 then uses the
properties computed above to determine the location of feature
edges of the 3D part/mold being modeled. A feature edge is an edge
that would be apparent to one viewing the part/mold. The feature
edge locator 612 identifies elements of the surface mesh adjacent
to a feature edge. In one embodiment, the feature edge locator
identifies feature edges by first classifying each of the elements
of the surface mesh as planar elements or "curved" elements. For
example, a planar (triangular) element either (1) has each of its
three bend angles either equal to zero or greater than a given
feature edge threshold, or (2) has at least one adjacent planar
element where the bending angle between the element and the
adjacent planar element is zero. All other elements are classified
as "curved" elements (even though, individually, they are actually
planar). In one embodiment, the feature edge locator 612 in FIG. 6
identifies feature edges at the following locations: (1) where the
bend angle between two adjacent elements is greater than a given
feature edge threshold (for example, from about 40.degree. to about
45.degree.); (2) at the edge between planar elements and curved
elements; and (3) where there is a significant change in bending
(element) curvature direction, for example, as shown in the diagram
800 of FIG. 8B.
[0090] Once the feature edge locator has located feature edges,
further organization of the subsurfaces is performed to identify
the remaining edges separating all of the subsurfaces of the
surface mesh. The subsurface classifier 616 in FIG. 6 performs an
element-by-element, pseudo-recursive process to group elements into
subsurfaces according to bending angle and bending curvature, so
that adjacent elements having similar bending curvature are grouped
together. High-curvature subsurfaces will generally bound regions
of the 3D-analysis portion of the solution domain, while planar and
low-curvature subsurfaces will generally bound regions of the
2.5D-analysis portion of the solution domain. The pseudo-recursive
process performed by the subsurface classifier 616 begins by
determining the location of large planar subsurfaces (sheets) of
the surface mesh. In one embodiment, a large planar sheet is a
planar sheet in which one of its elements has an area greater than
a threshold value, based on the mesh geometry size, average
associated thickness, and number of elements. Here, the size of the
elements are based on a reasonable aspect ratio. By identifying
large planar sheets first, the subsurface classifier 616 avoids
grouping large elements into curved subsurfaces. After a large
planar sheet is identified, adjacent planar elements are added to
the large planar sheet as long as planar surface (sheet)
constraints are met. In one embodiment, the planar sheet
constraints are as follows:
[0091] (1) The edge bend angle inside the sheet (i.e., the maximum
element-to-element bend angle along the edge, as illustrated in
FIGS. 7A and 7B) is less than a tolerance value, set, for example,
from about 5.degree. to about 15.degree.; and
[0092] (2) Each node of the sheet has an off-distance below a
tolerance level, set from about 0.05 times an average associated
thickness (see below) to about 0.1 times the average associated
thickness;
[0093] where "off-distance" is the perpendicular distance from a
node to the "plane of the subsurface," and the plane of the
subsurface is the plane defined by the largest element of the
subsurface.
[0094] The next step in the subsurface classifier 616 is the
identification of "other" (not large) planar sheets. In one
embodiment, all connecting planar (non-"curved") elements that are
not already part of a large planar sheet make up one of these
"other" planar sheets.
[0095] The next step in the subsurface classifier 616 is the
element-by-element, pseudo-recursive classification of
low-curvature subsurfaces, followed by high-curvature subsurfaces.
Adjacent "curved" elements (as defined above) with similar
curvatures are grouped into an individual curved sheet
(subsurface). The pseudo-recursive process proceeds by applying
criteria to determine whether an adjacent "curved" element belongs
to the current curved subsurface. In one embodiment, there are four
criteria used to determine if a neighboring (adjacent) element
belongs to the current curved subsurface:
[0096] (1) The bending angle between the neighboring (candidate)
element and the current subsurface is less than about
1.degree.;
[0097] (2) The curvature of the neighboring element is less than a
threshold value (applied for low-curvature subsurfaces). For
example, a low-curvature subsurface must have a maximum curvature
less than or equal to about (0.06/thickness), where "thickness" is
the average thickness associated with the subsurface (see
below);
[0098] (3) The curvature of the current subsurface and the
neighboring element is larger than about (0.5/thickness), where
"thickness" is the average thickness associated with the current
subsurface (applied for high-curvature subsurfaces); and
[0099] (4) The bending angle between the neighboring element and
the current subsurface is less than a threshold edge bending angle
(set, for example, from between about 30.degree. and about
45.degree.), and the curvature of the neighboring element is less
than about 4 times the minimum curvature of the current subsurface,
where the minimum curvature of the subsurface is the smallest
element-to-element bend angle of all elements belonging to the
subsurface.
[0100] The next step in the subsurface classifier 616 is to group
all the remaining elements into planar subsurfaces. The subsurface
classifier 616 then identifies the final edges separating the
subsurfaces, and computes and stores the following properties for
each edge:
[0101] Length;
[0102] Bending angle;
[0103] Direction of bending (in or out); and
[0104] Adjacent elements.
[0105] Next, the subsurface classifier 616 identifies surface
loops. Surface loops are the oriented edges of the subsurfaces. For
example, a rectangular surface with a hole cut in it will have two
associated loops--one for the outer edges of the rectangle and one
describing the interior hole. The subsurface classifier 616
computes and stores the following loop properties:
[0106] Length; and
[0107] Edges connected to the loop.
[0108] Finally, the subsurface classifier 616 computes and stores
the following properties for each subsurface:
[0109] Perimeter;
[0110] Area;
[0111] Nodes in the subsurface;
[0112] Elements in the subsurface;
[0113] Edges;
[0114] Loops; and
[0115] Minimum measure of curvature associated with the
subsurface.
[0116] Subsurface curvature, as described herein, is different from
element curvature in that subsurface curvature is characterized by
a minimum, maximum, average, and/or range of the element edge
curvatures belonging to the subsurface.
[0117] Once the subsurfaces are classified, the mesh associated
with each subsurface is further refined or coarsened according to
given criteria for optimizing mesh quality and efficiency, for
purposes of numerical analysis. For example, the remesher 620 in
FIG. 6 coarsens or refines the mesh of a given subsurface according
to a user-defined value of Specified Edge Length (SEL). The larger
the value of SEL, the coarser the remeshed mesh will be and,
conversely, the smaller the value of SEL, the finer the remeshed
mesh will be. In one embodiment, a default value of SEL is
calculated based on the complexity of the model. A user can
increase or decrease the value of SEL to trade off accuracy for
analysis speed and, vice versa, subject to internal constraints. In
one embodiment, internal constraints are imposed on allowable
values of SEL such that the maximum off-distance of the remeshed
nodes to their positions in the original mesh is below a given
threshold (for example, from about 1% to about 5% of SEL), and such
that the maximum bend angle is below a given threshold (for
example, a value from about 15.degree. to about 30.degree.).
[0118] In one embodiment, the remesher 620 in FIG. 6 performs the
following sequence:
[0119] (1) After all the subsurfaces of the surface mesh are
classified by the subsurface classifier 616, the boundaries of the
subsurfaces (edges) are remeshed. Here, the loops of each sheet are
divided into lengths (SEL) by inserting and/or merging nodes on the
current set of edges that define the loops;
[0120] (2) Constraints are applied in the merging of edge nodes
according to internal constraints on SEL. These constraints ensure
that the loop does not drift substantially from its initial shape.
For example:
[0121] (a) Bend angle constraints are applied: If the loop is bent
more than a specified bend angle, then that section of the loop is
not subject to the edge length criteria (edge nodes should not be
merged according to SEL).
[0122] (b) Chord height constraints are applied: If the chord
height of a short edge node (a node on an edge shorter than SEL)
with respect to its adjacent node is larger than a specified
length, then that section of the loop is not subject to the edge
length criteria (edge nodes should not be merged according to SEL);
and
[0123] (3) SEL is applied to the mesh of the subsurface to be
remeshed. For example, an iterative "bisection and merge algorithm"
is performed as follows:
[0124] (a) Start with the longest element in the subsurface and
insert nodes by bisecting the longest edge of the element if the
edge is significantly larger than SEL (for example, if the edge is
larger than about 1.5 times SEL).
[0125] (b) Look for element edges that are significantly shorter
than SEL (for example, where the edge is smaller than 0.5 times
SEL), and merge these nodes together.
[0126] (c) Repeat until all the element edge lengths lie in a close
range of SEL.
[0127] FIG. 9 depicts the application of "bend angle constraints"
and "chord height" constraints in item (2) above. The chord height
908 must be less than a given threshold, and the bend angle must be
less than a given threshold in order for SEL to be applied in
coarsening a mesh by merging original nodes 902 on the loop to form
new elements 906. FIG. 9 shows seven original elements 904 which
are replaced by two new elements 906 following merging according to
a value of SEL, subject to bend angle constraints and chord height
constraints.
[0128] Note that in all of the mesh modification operations, the
element node linkages can be modified to optimize the aspect ratio
about a node at any time in the process. This is a local
optimization operation, and can be defined as one or more "rules"
for meshing around nodes. This causes certain components of the
system represented in FIG. 6 to be iterative and/or recursive.
[0129] FIG. 10 is a schematic 1000 that demonstrates the formation
of additional elements on a subsurface 1002 at sequential stages of
a remeshing procedure, as in the "bisection and merge" algorithm in
item (3) above, or as in a local optimization operation. In this
case, the mesh of a subsurface is further refined by bisecting the
longest side of an element on the subsurface to create additional
elements. For example, mesh 1002 is the initial mesh. The mesh is
refined by defining a node at the midpoint of the longest element
side and extending lines to one or more vertices not yet connected
to the midpoint, thereby creating one or more additional triangles.
This is illustrated by subsequent meshes 1004, 1006, 1008, 1010,
and 1012 in FIG. 10. In meshes 1004, 1006, 1008, 1010, and 1012,
the midpoint of the longest side (denoted by "o") and the dotted
lines extending from this point define the new elements. Remeshing
continues until the elements satisfy one or more criteria on size
(i.e., given by SEL). For example, FIG. 11 shows an initial
stereolithography surface representation 1100 used in a remeshing
procedure. FIG. 12 shows the surface representation of FIG. 11
after remeshing (1200) using the bisection algorithm discussed
above.
[0130] After the subsurfaces are remeshed, the subsurface matcher
624 in FIG. 6 determines which subsurfaces are "matched
subsurfaces," and the remaining subsurfaces are each classified as
"unmatched subsurfaces" or "edge subsurfaces." The matched
subsurfaces are later connected with wedge elements to form a first
portion of the solution domain--the portion in which 2.5D analysis
can be accurately performed.
[0131] Matched subsurfaces are those that are related to another
surface such that a notion of thickness between them can be
sensibly defined. The thickness between matched subsurfaces is
either substantially invariant or gradually varying. For example,
FIG. 13 depicts a cross-section 1300 of a T-shaped object (a
filleted rib) and illustrates matching individual subsurfaces of
the overall surface representation. In the cross-section 1300, line
segments ab, cd, and gh are on edge subsurfaces. Line segments aj
and ed are matched to bc. Line segment fg is matched to hi. The
curved sections ij and ef are unmatched. It is not possible to
sensibly define a thickness of unmatched subsurfaces ij and ef.
[0132] FIG. 14 further illustrates the concept of matching
subsurfaces. FIG. 14 shows a cross section 1400 through a filleted
rib. Here, line segments ab, cd, and gh are on edge subsurfaces.
Line segments aj and ed are matched to bc. Line segments fg and hi
are matched. Curved sections ij and ef are unmatched. Note that
line segments fg and hi are still considered matched, despite the
taper. However, if the taper is extreme, the surfaces forming the
tapered ribs may not be matched.
[0133] Matched subsurfaces are subsurfaces containing matched
elements. Matching is performed element-by-element and
subsurface-by-subsurface until all the elements that can be matched
are considered. In one embodiment, the subsurface matcher 624 of
FIG. 6 applies criteria to determine whether two triangular
elements are matched or not. In one embodiment, these two criteria
are (1) whether the "Triangle Match Angle" (TMA) is less than a
given value (for example, from about 30.degree. to about
45.degree.) and (2) whether "Triangle Match Distance" (TMD) is less
than a given value based on the mesh average thickness, the
matching subsurface's average width, and the matching subsurface's
boundary characteristic. A boundary characteristic of a subsurface
is the ratio of its expending edge length to the total edge length.
The determination of TMA and TMD are demonstrated in the diagram
1500 of FIG. 15. To determine TMA, first, find the centroid of the
triangle to be matched (Triangle A in FIG. 15). Then, project it
along its normal to find the intersection with a triangle on a
subsurface on the opposite side of the model (surface
representation). Here, this is the "Projection point" on Triangle B
in FIG. 15. TMA is then calculated as shown in FIG. 15. Triangle
Match Distance, TMD, is the perpendicular distance of the
projection used in calculating TMA, shown in FIG. 15.
[0134] The subsurface matcher 624 in FIG. 6 uses a "collapse"
procedure to identify which of the unmatched subsurfaces are edge
subsurfaces, and to assign the following subsurface properties: (1)
a sheet (subsurface) type, indicating how it was collapsed
(primary/secondary or edge); (2) a "move distance" for each node;
and (3) a "move vector" for each node.
[0135] FIGS. 16A through 16C demonstrate steps in an example
collapse procedure. Matched subsurfaces 1602 and 1604 are
"collapsed" together to a final collapse position 1608, shown in
diagram 1600 of FIGS. 16A and 16B. The collapse direction is
established using the matching information from the previous mesh
step and the subsurface properties, including the area,
eigendimension, and the boundary characteristic. "Primary sheets"
are the subsurfaces that are chosen to be moved first. The moving
distance of a node on a primary sheet provides the average matching
distance of the sheet. The moving distance of a node on a secondary
sheet is the distance from the node to the opposite primary sheet
minus the primary sheet moving distance. After collapse, the nodes
on both the primary 1602 and secondary sheets 1604 in FIG. 16A have
been moved to the midplane 1608. Non-matched sheets (subsurfaces)
(1642, 1644 of FIG. 16C) do not move in the collapsing process. A
non-matching sheet whose area shrinks to about 20% or less of its
original size after primary and secondary sheets are collapsed to
the midplane is considered an "edge subsurface." Subsurfaces 1642
and 1644 in FIG. 16 are edge subsurfaces, since their areas shrink
to zero (or near zero), in the example shown in FIG. 16. Note that
the collapse procedure is used to identify edge surfaces and to
assign subsurface properties. The collapse procedure is not used to
consolidate the mesh into a midplane representation. The mesh is
"re-inflated" to its original node positions, shown in diagram 1640
of FIG. 16C.
[0136] After categorizing the subsurfaces, the subsurface matcher
624 of FIG. 6 assigns thicknesses to the subsurfaces. These
thicknesses are used, for example, in performing a 2.5D flow
analysis (i.e. using a Hele Shaw approximation) in a first portion
of the solution domain. The subsurface matcher 624 assigns a
thickness to elements on matched subsurfaces equal to the average
distance between the subsurfaces. Elements on an edge subsurface
are assigned the thickness of the matched subsurfaces to which they
are attached. Elements on unmatched subsurfaces are assigned an
average thickness of surrounding elements on matched subsurfaces.
In some embodiments, thicknesses are not assigned to edge
subsurfaces and/or unmatched subsurfaces, as they may be
unnecessary for performing a flow analysis in certain
applications.
[0137] The subsurface matcher 624 categorizes each of the
subsurfaces of the model as either a matched, unmatched, or edge
subsurface, and determines a set of paired elements that define the
possible 2.5D-analysis portion of the solution domain between the
matched subsurfaces (the 2.5D-analysis portion may comprise one or
more non-contiguous regions). After the possible 2.5D regions are
identified, the final 2.5D regions making up the 2.5D-analysis
portion of the solution domain are determined by removing all
paired elements which: (1) connect to surface edges for which the
edge bending angle is greater than a given value (for example,
about 30.degree.); (2) connect to unmatched elements that do not
belong to an edge surface; and/or (3) form a small patch of 2.5D
regions. The remaining paired elements define (bound) the
2.5D-analysis portion of the solution domain (first portion), and
the rest of the elements define (bound) the 3D-analysis portion of
the solution domain (second portion).
[0138] After the solution domain is divided into a 2.5D-analysis
portion and a 3D-analysis portion, the portions are discretized
using the matched, unmatched, and edge subsurface elements. In one
embodiment, the element pairs in the 2.5D-analysis portion are
converted into 6-node wedge elements, and the remaining subsurface
elements are closed up with triangular elements to form the
3D-analysis regions, which are meshed with tetrahedral
elements.
[0139] The first portion solution domain discretizer 628 in FIG. 6
converts the matched element pairs that define the 2.5D-analysis
portion (first portion) into 6-node wedge elements (where each
wedge element has one node at each of its six corners).
Calculations are performed at each grid point. The six nodes
defining the wedge element may all have different pressures, and
solution grid points through the thickness of the wedge may provide
increased accuracy. However, in thin areas where the flow
approaches two-dimensional flow, the pressures on the top three
nodes of the wedge are about the same as the pressures on the
respective bottom three nodes of the wedge, and either no grid
points are needed, or only 1, 2, or 3 grid points are needed, for
example. In one example, the number of grid points used along the
thickness of an element is in a range from about 3 to about 40. In
another embodiment, the number of grid points used along the
thickness of an element is in a range from about 8 to about 20. A
lower number of grid points provides a faster analysis, which may
be more suitable for design iterations early in the development of
a product. For each wedge element, there may be any number of grid
points through the thickness of the element, depending on the
desired accuracy of the model. In some embodiments, there are no
solution grid points between element nodes, and solutions are
obtained only at the six nodes of each element. With the addition
of grid points, each wedge is a discretely layered element. In an
alternative embodiment, each wedge is a continuously layered
element.
[0140] The interface element locator 632 in FIG. 6 defines
interface elements along the boundary between the first portion
(i.e. 2.5D-analysis portion) and the second portion (i.e.
3D-analysis portion) of the solution domain. The interface element
locator 632 uses one or more of the following types of interface
elements, for example, according to the types of first portion and
second portion elements to be linked:
[0141] Disk-shaped element with a central node on the end of a
1D-analysis element and a plurality of surrounding nodes belonging
to the 3D-analysis region elements which contact the end face of
the 1D-analysis element;
[0142] Line-shaped element which connects the 2.5D-analysis element
(i.e. wedges) and 3D-analysis elements (i.e. tetrahedra);
[0143] Rectangular-shaped element with 4 corner nodes belonging to
a 2.5D-analysis element (which may or may not connect to
3D-analysis elements) and any number of 3D-analysis element nodes
lying on and inside that 2.5D-analysis element; and
[0144] Triangular-shaped element with 3 corner nodes belonging to a
2.5D-analysis element (which may or may not connect to 3D-analysis
elements) and any number of 3D-analysis element nodes lying on and
inside that 2.5D-analysis element.
[0145] Alternatively, the interface element locator 632 may use a
different type of element than those listed above.
[0146] In an embodiment in which the first portion of the solution
domain comprises wedge elements, a set of line elements is created
along the interface of the 2.5D-analysis portion and the
3D-analysis portion after the 2.5D-analysis portion is meshed with
wedge elements by using nodes at the corners of each wedge plus one
or more grid point nodes in between. The number of grid points used
may be from about 3 to about 40. Generally, the number of grid
points ranges from about 8 to about 20. Alternatively, fewer (0, 1,
or 2) or more (over 40) grid points than indicated by these ranges
is used.
[0147] The interface element locator 632 uses all of the nodes of
the interface elements to make triangular elements to close the
3D-analysis portion (second portion) of the solution domain. At the
open edges of the 2.5D-analysis portion (first portion), the grid
points and nodes forming the wedges are discretized with a surface
mesh to ensure that the first and second portions are connected.
For example, in applying the above classification and
discretization procedure to a planar, thin square plate, the region
to be meshed with 2.5D-analysis wedges (first portion of the
solution domain) is defined internal to all edges of the plate. The
region between the wedges and the exterior edges of the plate are
then meshed with 3D-analysis tetrahedral elements. This mesh allows
accurate calculation of heat loss at the edge of the plate.
However, for thin regions, the heat loss is minimal and may be
ignored. Thus, in one example, the invention automatically places
wedge elements at free edges of the model in order to lower the
number of tetrahedral elements needed.
[0148] After the interface elements are located and the closing
step above is performed, the second portion solution domain
discretizer 636 in FIG. 6 discretizes the 3D-analysis portion with
three-dimensional elements. The three-dimensional elements can be
tetrahedral elements, hexahedral elements, or some combination of
the two. However, any type or combination of polyhedral elements
can be used.
[0149] In an alternate embodiment, the steps of discretizing the
first and second portions of the solution domain and creating
interface elements are ordered differently than described above.
For example, components 628, 632, and 636 of the system of FIG. 6
may operate in a different order than shown in FIG. 6. For
instance, the relative ordering of these three components of the
system of FIG. 6 may be any of the following: (1) 628, 632, 636;
(2) 628, 636, 632; (3) 632, 628, 636; (4) 632, 636, 628; (5) 636,
628, 632; and (6) 636, 632, 628.
[0150] FIG. 17 depicts an example of a hybrid mesh solution domain
1700 with a 2.5D-analysis portion and a 3D-analysis portion
separated by interface elements, automatically created from CAD
output according to an embodiment of the invention. The
2.5D-analysis portion is made up of regions 1702, 1704, 1706 (light
colored) that are discretized with wedge elements. The wedge
elements connect matched subsurfaces of the model. The 3D-analysis
portion 1708 (darker colored) is discretized with tetrahedral
elements. Linear interface elements lie along the boundary of the
2.5D-analysis and 3D-analysis portions and are shown as heavy line
segments, such as those at 1710 and 1712. FIG. 18 depicts a
close-up 1800 of the interface elements (for example, 1710, 1712,
1802, 1804, 1806, 1808, 1810) and tetrahedral elements (1708) of
the hybrid solution domain in FIG. 17.
[0151] The equation solver 642 in the system 600 of FIG. 6 solves
for the distribution of one or more process variables (such as
pressure, temperature, flow velocity, stress, viscosity, and fluid
flow front) in the first and second portions of the solution domain
as functions of time. The governing equations include mass,
momentum, and energy balances, and they are solved (concurrently)
in the respective portions of the solution domain, subject to
process inputs 640 that describe the process being modeled, initial
conditions, and boundary conditions. Examples of solution
procedures performed by the equation solver 642 are shown herein
below for the 2.5D-analysis portion of the solution domain, the
3D-analysis portion of the solution domain, and the interface
elements.
[0152] The 2.5D-analysis portion of the solution domain may be
discretized using wedge elements that have or do not have grid
points along their thicknesses. A low Reynolds number fluid flow is
typical for fluid injection into narrow regions such as those that
make up the 2.5D-analysis portion (first portion) of the solution
domain. In one embodiment, a general Hele-Shaw approximation is
used for process simulation with low Reynolds number flow in the
2.5D-analysis portion. The governing equations include momentum,
energy, and mass (continuity) conservation equations, and are
applied in the 2.5D-analysis portion of the solution domain. The
governing equations for the 2.5D-analysis portion are shown in
Equation 2 through Equation 4 as follows: 25 P x = z ( x z ) ( 2 )
P y = z ( y z ) P z = z ( z z ) c p ( T t + -> T ) = 2 + ( T ) (
3 ) t + ( x x + y y + z z ) = 0 ( 4 )
[0153] where Equation 2 represents the conservation of momentum
equation in Cartesian coordinates (z is the thickness direction),
Equation 3 represents the conservation of energy equation, and
Equation 4 represents the continuity (conservation of mass)
equation. In one embodiment, equations 2-4 are solved in each
region of the 2.5D-analysis portion subject to the boundary
conditions shown in Equations 5 and 6 as follows:
{overscore (.upsilon.)}(z=.+-.h)=0 (5)
T(z=.+-.h)=T.sub.W (6)
[0154] where T.sub.W is the mold wall temperature, and the mold
walls are located at z=h and z=-h, where h is the halfwall
thickness associated with the given region of the 2.5D-analysis
portion. Heat conduction in the x- and y-directions may be ignored,
and a slab formulation may be used to facilitate the calculation of
temperature profile and viscosity profile (where viscosity may be a
strong function of temperature). In an alternative embodiment, this
simplification is not made. A finite difference method may be used
for the solution of the energy balance in the 2.5D-analysis
portion, where convection is based on an up-winding scheme. An
example of an upwinding scheme is described in co-owned European
Patent Number 1218163, issued Nov. 19, 2003, and U.S. patent
application Ser. No. 09/404,932, the disclosures of which are
incorporated herein by reference in their entirety.
[0155] By combining Equation 2 (momentum balance) and Equation 4
(continuity equation), the equation solver 642 in FIG. 6 derives
finite element equations for the pressure field in the
2.5D-analysis portion of the solution domain using a Galerkin
weighted residual approach. Equation 7 is obtained by expressing
lumped mass on the right side of the equation: 26 S e ( A e T + A e
B ) N i N j p i = S e L ij p i = V n j e j t ( 7 )
[0156] where S.sub.e is the elemental flow conductance, defined as
in Equation 8: 27 S e = 0 h z 2 ( z ) z ( 8 )
[0157] and where A.sub.e.sup.T is the top area of an element and
A.sub.e.sup.B is the bottom area of the element. In general, an
asymmetric temperature profile results in an asymmetric viscosity
profile. In this case, the flow conductance in a slab channel may
be expressed as in Equation 9: 28 S e = 1 2 { h - h + z 2 ( z ) z -
( h - h + z ( z ) z ) 2 h - h + 1 ( z ) z } ( 9 )
[0158] where integrals are evaluated from z=-h to z=h.
[0159] Thus, distributions of any of the following process
variables throughout the 2.5D-analysis portion may be obtained as
functions of time, for example: temperature, pressure, fluid
velocity, fluid flow front position, internal energy, density,
fluidity, viscosity, and gradients thereof.
[0160] In an alternative embodiment, the distribution of a process
variable throughout the 2.5D-analysis portion of the solution
domain is determined using the method of U.S. Pat. No. 6,096,088 to
Yu et al., the disclosure of which is incorporated herein by
reference in its entirety, so that flow fronts along matching
subsurfaces are synchronized. In an embodiment employing this
solution technique for the 2.5D-analysis portion of the solution
domain, interface elements are planar in shape and lie between the
2.5D-analysis regions and the 3D-analysis regions.
[0161] The 3D-analysis portion of the solution domain is
discretized with three-dimensional tetrahedral elements; however,
other shapes may be used. The 3D analysis may include solution of
Navier Stokes equations or the simplified Stokes equation, where
inertia and gravity are ignored. Body forces such as inertia and
gravity are generally negligible in injection molding where the
fluid has a high viscosity and a low Reynolds number, but this
simplification is not necessary.
[0162] The governing equations that are solved in the 3D-analysis
portion include momentum, energy, and mass (continuity)
conservation equations. In one embodiment, the generalized momentum
equation is expressed as in Equation 10: 29 -> t = g -> - P +
[ . ] - [ -> -> ] ( 10 )
[0163] Assuming negligible body forces, the momentum equations are
expressed by the Stokes equation, Equation 11:
[.gradient..multidot..eta.{dot over (.gamma.)}]-.gradient.P=0
(11)
[0164] In Cartesian coordinates, the Stokes equation is expressed
as in Equation 12: 30 P x = x ( 2 x x ) + y ( ( y x + x y ) ) + z (
( z x + x z ) ) P y = x ( ( y x + x y ) ) + y ( 2 y y ) + z ( ( z y
+ y z ) ) P z = x ( ( z x + x z ) ) + y ( ( y z + z y ) ) + z ( 2 z
z ) ( 12 )
[0165] The continuity (mass conservation) equation in the
3D-analysis region is expressed as in Equation 13: 31 t + -> +
-> = 0 ( 13 )
[0166] For modeling an injection molding process, the following
boundary condition in Equation 14 may be applied:
{overscore (.nu.)}({overscore (x)})=0, .A-inverted.{overscore
(x)}.epsilon..differential..OMEGA. (14)
[0167] where .differential..OMEGA. is the mold/plastic interface.
For linear tetrahedral elements in the 3D-analysis portion of the
solution domain, the equation solver 642 uses element interpolation
functions as the weighting functions. Applying the Bubnov-Galerkin
approach then yields residual Equations 15-22: 32 R x = x ( 2 x x )
+ y ( ( y x + x y ) ) + ( 15 ) z ( ( z x + x z ) ) - P x ; R x , N
i e = 0 ( 16 ) = V e [ ( 2 x x - P ) N i x + ( y x + x y ) N i y +
( z x + x z ) N i z ] V - S e _ x N i S R y = x ( ( y x + x y ) ) +
y ( 2 y y ) + ( 17 ) z ( ( z y + y z ) ) - P y ; R y , N i e = 0 (
18 ) = V e [ ( y x + x y ) N i x + ( 2 y y - P ) N i y + ( z y + y
z ) N i z ] V - S e _ y N i S R z = x ( ( z x + x z ) ) + y ( ( y z
+ z y ) ) + ( 19 ) z ( 2 z z ) - P z ; R z , N i e = 0 ( 20 ) = V e
[ ( z x + x z ) N i x + ( z y + y z ) N i y + ( 2 z z - P ) N i z ]
V - S e _ z N i S R P = t + -> + -> ; ( 21 ) R P , N i e = 0
= V e ( t + -> + -> ) N i v ( 22 )
[0168] Using linear interpolation functions to approximate both
velocities and pressure in the tetrahedral element, the elemental
stiffness matrix in Equation 23 results: 33 [ 2 k _ _ 11 + k _ _ 22
+ k _ _ 33 k _ _ 12 k _ _ 13 L _ _ 1 k _ _ 21 k _ _ 11 + 2 k _ _ 22
+ k _ _ 33 k _ _ 23 L _ _ 2 k _ _ 31 k _ _ 32 k _ _ 11 + k _ _ 22 +
2 k _ _ 33 L _ _ 3 L _ _ 1 T L _ _ 2 T L _ _ 3 T 0 _ _ ] { _ x _ y
_ z P _ } = { R _ x R _ y R _ z 0 _ } or [ K _ _ L _ _ L _ _ T 0 _
_ ] { -> _ P _ } = { R -> _ 0 _ } for short . ( k _ _ ij ) kl
= V e N k x i N l x j V , ( L _ _ i ) kl = - V e N k x i N l V = -
N k x i V e 4 , ( 23 )
[0169] Without modification, this system may be ill-posed, since it
does not satisfy the "inf-sup" or Babuska-Brezzi stability
condition. Spurious pressure modes may cause severe oscillation in
the pressure solution, and the velocity solution may lock,
regardless of mesh size. Therefore, the Equation solver uses a
"Mini" element formulation to stabilize the system. In the Mini
element formulation, an enriched space of velocity trial functions
is constructed out of the linear trial space and the space of
bubble functions as in Equation 24:
{overscore (.upsilon.)}.sub.+={overscore
(.upsilon.)}.sub.l+{overscore (.upsilon.)}.sub.b (24)
[0170] where {overscore (.upsilon.)}.sub.l is the usual linear
interpolation in the element and {overscore (.upsilon.)}.sub.b is
the bubble velocity in the element. The bubble velocity is
expressed in terms of a bubble shape function, .phi.({overscore
(x)}), as in Equation 25: 34 b = ( x ) b e { ( x ) > 0 x e \ e (
x ) = 0 x e ( x ) = 1 x = geometrical center of e ( 25 )
[0171] The quantity {overscore (.upsilon.)}.sub.b.sup.e is an
element vector such that {overscore (.upsilon.)}.sub.b has constant
direction in an element but a varying magnitude determined by the
bubble shape function, .phi.({overscore (x)}) . A cubic bubble
shape function, which is actually quartic in three-dimensions, is
one option, shown as in Equation 26:
.phi.({overscore (x)})=4.sup.4N.sub.1({overscore
(x)})N.sub.2({overscore (x)})N.sub.3({overscore
(x)})N.sub.4({overscore (x)}), .A-inverted.{overscore
(x)}.epsilon.e (26)
[0172] A quadratic bubble shape function may be used for greater
stability, as in Equation 27: 35 ( x ) = 2 - 4 k ( N k ( x ) ) 2 ,
x e ( 27 )
[0173] Since the linear subspace and the bubble subspace are
orthogonal, Equation 28 applies:
.intg..sub.e.gradient.{overscore
(.upsilon.)}.sub.l:.gradient.{overscore (.upsilon.)}.sub.bdV=0
(28)
[0174] where V.sup.e is element volume. Substituting Equations 24
through 28 into Equation 23 produces a linear system of equations
with the structure shown in Equation 29: 36 [ K _ _ 0 _ _ L _ _ 0 _
_ K _ _ b L _ _ b L _ _ T L _ _ b T 0 _ _ ] { _ l b P _ } = { R _ R
b 0 _ } K _ _ b = [ 2 k 11 b + k 22 b + k 33 b k 12 b k 13 b k 21 b
k 11 b + 2 k 22 b + k 33 b k 23 b k 31 b k 32 b k 11 b + k 22 b + 2
k 33 b ] ( 29 )
[0175] where, for a quadratic bubble function, Equations 30 and 31
apply: 37 k ij b = V e x i x j V = 16 5 V e e k = 1 4 l = k 4 N k x
i N l x j , ( 30 ) ( L _ _ b ) ij = V e x i N j V = 1.6 V e m = 1 4
( 2 - jm ) N m x i , ( 31 )
[0176] and, for a cubic bubble function, the following applies: 38
k ij b = V e x i x j V = 4096 945 V e e k = 1 4 l = k 4 N k x i N l
x j , ( 32 ) ( L _ _ b ) ij = V e x i N j V = 32 V e 105 m = 1 4 (
2 - jm ) N m x i . ( 33 )
[0177] Since the bubble velocities in this embodiment are only
defined inside the element under consideration, the system in
Equation 29 is reduced by static condensation of the bubble
velocities to produce Equation 34: 39 [ K _ _ L _ _ L _ _ T - L _ _
b T K _ _ b - 1 L _ _ b ] { _ l P _ } = { R _ - L _ _ b T K _ _ b -
1 R _ _ b } ( 34 )
[0178] Having solved for the linear part of the velocity,
{overscore (.upsilon.)}.sub.l, the actual velocity, {overscore
(.upsilon.)}.sub.+, is obtained from Equation 24. For a cubic
bubble, the bubble velocity is zero at nodes such that the nodal
values of {overscore (.upsilon.)}.sub.l are, in fact, the desired
solution. For a quadratic bubble, the bubble velocity at each node
within an element, according to Equation 27, is -2{overscore
(.upsilon.)}.sub.b.sup.e. In one embodiment, this term is
considered to be negligible.
[0179] Alternatively, the equation solver 642 of FIG. 6 solves for
the distribution of process variables in the 3D-analysis portion of
the solution domain using a boundary element method (BEM). Here,
the boundary element method only requires an external mesh and
there is no need to mesh the interior of the 3D-analysis portion of
the solution. A boundary element method may be applied for simple
fluids and/or for linear structural analysis. The equation solver
642 may solve a nonlinear problem with an extended BEM or a
meshless technique by inserting points within the 3D-analysis
portion.
[0180] The equation solver 642 in FIG. 6 can solve for the
temperature field in the 3D-analysis portion of the solution
domain. The generalized energy conservation equation that is solved
in the 3D-analysis portion is shown in Equation 35: 40 c p ( T t +
T ) = T ( P t + P ) + . 2 + ( T ) + Q . ( 35 )
[0181] Equation 35 accounts for the variation of temperature in a
mold as a function of position and time, due to convection,
compressive heating, viscous dissipation, heat conduction to/from
the mold, and/or heat sources such as heat of reaction and/or other
heat source effects. The energy conservation equation is generally
solved concurrently with the mass and momentum conservation
equations. Equation 35 may be solved using one or more finite
element techniques, finite different techniques, or a combination
of finite difference and finite element techniques.
[0182] In one example, the equation solver 642 solves the energy
balance of Equation 35 for the 3D-analysis portion of the solution
domain using a finite element method. Shear heating and compressive
heating may be explicitly calculated based on the results of the
preceding time step. Convection may be calculated based on an
up-winding method and temperature interpolation. Heat capacity can
be lumped or consistent. An example of an up-winding method and
temperature interpolation method is described in co-owned European
Patent No. 1218163, issued Nov. 19, 2003, and U.S. patent
application Ser. No. 09/404,932, the disclosures of which are
incorporated herein by reference in their entirety.
[0183] Thus, distributions of any of the following process
variables throughout the 3D-analysis portion may be obtained as
functions of time, for example: temperature, pressure, fluid
velocity, fluid flow front position, internal energy, density,
fluidity, viscosity, and gradients thereof.
[0184] An energy balance is generally not solved for the interface
elements; however, their connectivity information may be used for
the heat convection calculation when heat is convected between a
2.5D-analysis region and a 3D-analysis region.
[0185] The interface elements can bridge up the geometry and/or
degrees-of-freedom discontinuities on the boundaries between
regions of the solution domain. For example, the equation solver
642 in FIG. 6 may treat the interface elements as general finite
elements, where an elemental matrix for an interface element is
described as in Equation 36:
k.sub.ij.sup.i=P*f(n.sub.i.sup.e,n.sub.j.sup.e) (36)
[0186] where P* is the "penalty number." In one example, the
equation solver 642 uses a penalty method to formulate the
elemental matrix to enforce continuity. Here, P* in Equation 36 is
a large number and f(n.sub.i.sup.e,n.sub.j.sup.e) is a function of
node i and node j of the given element. The elemental matrix of
interface elements for the pressure field may be expressed as in
Equation 37: 41 K ij = { - P f j T N j V N j i = N T , i j 0 i j ,
i , j N T , N B P N j V N j i = j , i , j N T , N B - P f j B N j V
N j i = N B , i j P f j T N j V N j i = j = N T P f j B N j V N j i
= j = N B f i T = ( z N l - h ) / 2 h , f i B = 1.0 - f i T N l = h
- h + z ' ( z ' ) z ' h - h + 1 ( z ' ) z ' h - z N l 1 ( z ' ) z '
- h - z N l 1 ( z ' ) z ' ( 37 )
[0187] where N.sub.T represents the top node and N.sub.B represents
the bottom node of the interface element, and where h is the
half-height of the interface element.
[0188] FIGS. 19-23 demonstrate the simulation of fluid flow within
an example mold cavity. The simulation includes automatically
creating a hybrid solution domain for the mold cavity using CAD
output, automatically discretizing the domain, and solving for the
distribution of process variables within the solution domain.
[0189] FIG. 19 depicts an injection-molded, plastic component 1900
for an automobile dashboard. In order to manufacture the plastic
component 1900, it is desired to create a model of the component
for performing a simulation of flow within the mold during
injection molding. The process simulation allows, for example,
adjustment of process conditions, injection point placement,
identification of potential processing trouble spots, and/or
adjustment of the component design at any stage of the design
and/or manufacturing process, without (or with a minimum of)
experimental trial-and-error.
[0190] FIG. 20 is a graphical representation 2000 of STL-formatted
CAD output produced during the design of the plastic component 1900
of FIG. 19. The mesh 2000 in FIG. 20 is not yet adapted for use in
finite element analysis. The system 600 depicted in FIG. 6 uses the
CAD output 602 to automatically create a hybrid solution domain
comprising a 2.5D-analysis portion and a 3D-analysis portion, which
is then used in finite element analysis. These steps are described
in more detail herein above.
[0191] FIG. 21 depicts the hybrid mesh solution domain 2100 for the
component 1900 as automatically determined using the system 600 of
FIG. 6. The hybrid mesh solution domain 2100 has two portions--the
light-colored portion 2104 is discretized using 6-node wedge
elements, and the dark-colored portion 2102 is discretized using
tetrahedral elements. A 2.5D flow analysis will be performed in the
light-colored portion 2104, and a 3D flow analysis will be
concurrently performed in the dark-colored portion 2102.
[0192] The flow analysis is performed, for example, by the equation
solver component 642 of the system 600 of FIG. 6, as described in
more detail above. The solver 642 determines the distribution of
process variables throughout the solution domain. For example, FIG.
22 depicts a map 2200 of times at which the flow front reaches
points within the mold of the component 1900 of FIG. 19, given the
location of the two injection points 2202 and 2204. The location of
the two injection points 2202 and 2204 are two of the process
inputs 640 used by the equation solver 642 in the system 600 of
FIG. 6. In the example shown in FIG. 22, the time required to
completely fill the mold for the component 1900 is 2.771 seconds.
According to the index at reference 2206, the red-colored portions
are the last portions of the mold to be filled.
[0193] FIG. 23 depicts the pressure distribution at the
filling/packing switchover point in the injection molding of the
component 1900 of FIG. 19. Pressure distribution is another of the
process variables determined using the hybrid mesh solution domain
2100 of FIG. 21 and the combined 2.5D/3D flow analysis approach
described herein. Other process variables that may be determined
include, for example, temperature distribution, fluid velocity,
viscosity, fluid flow front position, internal energy, density,
fluidity, and gradients thereof, all of which may be expressed as
functions of time.
[0194] FIG. 24 depicts a computer hardware apparatus 2400 suitable
for use in carrying out any of the methods described herein. The
apparatus 2400 may be a portable computer, a desktop computer, a
mainframe, or other suitable computer having the necessary
computational speed and accuracy to support the functionality
discussed herein. The computer 2400 typically includes one or more
central processing units 2402 for executing the instructions
contained in the software code which embraces one or more of the
methods described herein. Storage 2404, such as random access
memory and/or read-only memory, is provided for retaining the code,
either temporarily or permanently, as well as other operating
software required by the computer 2400. Permanent, non-volatile
read/write memory such as hard disks are typically used to store
the code, both during its use and idle time, and to store data
generated by the software. The software may include one or more
modules recorded on machine-readable media such as magnetic disks,
magnetic tape, CD-ROM, and semiconductor memory, for example.
Preferably, the machine-readable medium is resident within the
computer 2400. In alternative embodiments, the machine-readable
medium can be connected to the computer 2400 by a communication
link. For example, a user of the software may provide input data
via the internet, which is processed remotely by the computer 2400,
and then simulation output is sent to the user. In alternative
embodiments, one can substitute computer instructions in the form
of hardwired logic for software, or one can substitute firmware
(i.e., computer instructions recorded on devices such as PROMs,
EPROMs, EEPROMs, or the like) for software. The term
machine-readable instructions as used herein is intended to
encompass software, hardwired logic, firmware, object code, and the
like.
[0195] The computer 2400 is preferably a general purpose computer.
The computer 2400 can be, for example, an embedded computer, a
personal computer such as a laptop or desktop computer, a server,
or another type of computer that is capable of running the
software, issuing suitable control commands, and recording
information. The computer 2400 includes one or more input devices
2406, such as a keyboard and disk reader for receiving input such
as data and instructions from a user, and one or more output
devices 2408, such as a monitor or printer for providing simulation
results in graphical and other formats. Additionally, communication
buses and I/O ports may be provided to link all of the components
together and permit communication with other computers and computer
networks, as desired.
[0196] While the invention has been particularly shown and
described with reference to specific preferred embodiments, it
should be understood by those skilled in the art that various
changes in form and detail may be made therein without departing
from the spirit and scope of the invention as defined by the
appended claims.
* * * * *