Data processing: structural design – modeling – simulation – and em – Modeling by mathematical expression
Reexamination Certificate
2000-06-22
2004-04-06
Broda, Esq., Samuel (Department: 2123)
Data processing: structural design, modeling, simulation, and em
Modeling by mathematical expression
C703S005000, C703S014000, C716S030000
Reexamination Certificate
active
06718291
ABSTRACT:
FIELD OF THE INVENTION
The invention is directed to methods and apparata for solving problems of geometric modeling and engineering analysis, wherein the methods and apparata do not require spatial discretization of the underlying geometric domain or its boundary into elements which conform to the geometry (an activity commonly known as “meshing”).
BACKGROUND OF THE INVENTION
In engineering fields, geometric modeling of objects and the engineering analysis of the behavior of the modeled objects are extremely important activities. Modeling is generally performed by constructing a representation of an object's geometry on a computer (i.e., a CAD geometric model), with the representation including the “environment” of the object (that is, the object's boundary conditions, such as loads exerted on the object, temperatures on and around the object, and other physical and non-physical functional values). The analysis of the model's behavior is then usually also performed by computer, with the goal of predicting the modeled object's physical behavior based on the boundary conditions defined for the geometric model. In general, this involves determining physical functional values and/or their derivatives everywhere in the geometric model, both on its boundaries and in its interior—these values and derivatives being referred to herein as “field values”—based on the known boundary conditions defined at isolated locations on the model. The two activities of modeling and analysis are highly interrelated in that modeling is the prerequisite for analysis, while results of analysis are often used for further modeling. Most geometric models and analysis-related functions are represented in a piecewise fashion, which requires discretization of the model (e.g., construction of a mesh or grid) into finite elements that conform to and approximate the overall model.
However, modeling and analysis have largely emerged as two separate activities that are only weakly connected because they operate on distinct computer representations (e.g., the geometric domain of the model vs. the functional domain of the analysis), and the conversion process between these representations, effected by the aforementioned discretization, is time-consuming and difficult. This results in a slow and inefficient modeling and analysis cycle; the inability to reflect the results of analysis in the original model; difficulty in integrating multiple types of analyses on a common design model, and severely restricted types of analyses available for applications with time-varying geometries. Discretization (i.e., “meshing”) now dominates modeling and analysis activities both in manual effort and computer time. Thus, it is highly desirable to have “mesh-free” methods and apparata for modeling and analysis.
One of the main challenges for mesh-free modeling and analysis methods lies in constructing solutions to boundary value problems (e.g., differential equations) wherein the solutions satisfy the prescribed boundary conditions. The classical methods such as FEM (Finite Element Method), BEIM (Boundary Element Integral Methods), and FD (Finite Differences) rely on spatial discretization of the domain and/or its boundary in order to enforce or approximate the imposed boundary conditions at discrete locations. In contrast, the mesh-free methods discretize not the geometric domain but the underlying functional space—that is, the mathematical domain of the model rather than its physical domain. A number of mesh-free techniques which do not require discretization of the geometry have been developed (see T. Belytschko, Y. Krongauz, D. Organ, M. Fleming and P. Krysl, Meshless methods: An overview and recent developments.
Comput. Methods Appl. Mech. Engrg
., 139:3-47, 1996): smooth particle hydrodynamics (SPH); the diffuse element method (DEM); the reproducing kernel particle method (RKPM); the HP cloud method; the partition of unity method (PUM), and others. But geometric non-conformance of all such mesh-free methods (i.e., an inexact match between the geometric model and the functional model) makes treatment of boundary conditions problematic. Proposed remedies include the combination of the Element Free Galerkin Method (EFG) with finite element shape functions near the boundary; the use of modified variational principle; window or correction functions that vanish on the boundary; and Lagrange multipliers. Although these techniques appear promising for use in some cases, they often contradict the mesh-free nature of the approximation near the boundary, introduce additional constraints on solutions, or lead to systems with an increased number of unknowns (see, e.g. Frank C. Gunter and Wing Kam Liu, Implementation of boundary conditions for meshless methods.
Computer Methods in Applied Mechanics and Engineering
, 163:205-230, 1998).
Other mesh-free methods appear promising, but are cumbersome to use. In V. L. Rvachev,
Theory of R
-
functions and Some Applications
, Naukova Dumka, 1982 (in Russian), Rvachev developed the theory of R-functions—real-valued functions that behave as continuous analogs of logical Boolean functions. With R-functions, it became possible to construct functions with prescribed values and derivatives at specified locations, assisting in the solution of boundary value problems. Over the last several decades, the theory of R-functions has matured and has been applied to numerous scientific and engineering problems by Rvachev and his students, including problems of heat transfer, elasticity, magneto-hydrodynamics, various problems in inhomogeneous media, and many other areas. The theory of R-functions was implemented in a software system for scientific programming called POLYE (V. L. Rvachev and G. P. Manko,
Automation of Programming for Boundary Value Problems
, Naukova Dumka, 1983, in Russian; V. L. Rvachev, G. P. Manko, and A. N. Shevchenko, The R-function approach and software for the analysis of physical and mechanical fields, in J. P. Crestin and J. F. McWaters, editors,
Software for Discrete Manufacturing
, Paris, 1986, North-Holland). POLYE assists users in solving boundary value problems using a symbolic programming language called RL, in which a user may define a geometric domain, boundary conditions, and a solution structure for a selected problem. However, while POLYE and the RL language simplify the solution of boundary value problems, the problems are still largely manually solved insofar as the user must define the problem, including matters such as identifying and defining implicit functions for the geometry, selection and definition of solution structures, and the selection and definition of a solution procedure. Essentially, the POLYE software serves as an equation solver much like Mathematica (Wolfram Research, Champaign, Ill., USA), MATLAB (MathWorks, Inc., Natick, Mass., USA), or EES (Engineering Equation Solver, F-Chart Software, Middleton, Wis., USA), wherein the user is left to properly define the parameters of the problem and the software is specially configured to process the problem after it is properly defined. Therefore, unless the user has a high degree of knowledge and expertise in defining the problem parameters in the POLYE solver's language and environment, it is difficult to use. Other manual symbolic methods for construction of solution structures have been further explored in J. Kucwaj and J. Orkisz, Computer approach to the R-function method of solution of boundary value problems in arbitrary domains,
Computers
&
Structures
, 22(1):1-12, 1986.
Apart from the manual methods' need for significant experience and knowledge for their use, these methods also suffer from the significant drawback that they do not easily accommodate preexisting problem data, i.e., they do not allow the direct use of common engineering data such as CAD geometric models. Engineers often perform geometric modeling and problem definition in CAD systems because CAD systems are readily available and easy to use. They also allow the engineer to work in the geometric domain, w
Shapiro Vadim
Tsukanov Igor G.
Broda, Esq. Samuel
DeWitt Ross & Stevens S.C.
Fieschko, Esq. Craig A.
Phan Thai
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