Method and apparatus for analyzing engineering problems using a finite element technique with differential formulation

1. A computer-readable medium encoded with a computer program operable to perform a method comprising the steps of:

receiving input comprising a description of a thermally conductive physical object and a differential equation representing heat conduction of the physical object;

dividing a region into a plurality of finite elements to resolve an engineering problem, the region representing at least a portion of the physical object analyzed by the engineering problem;

determining for each of said finite elements a respective approximate solution for the differential equation which relates to said region, the approximate solution determined according to piecewise approximation of said region, the differential equation describing a physical property of the object and comprising one or more derivatives;

expressing each of the one or more derivatives according to a series expansion of each of the one or more derivatives to yield an expressed derivative;

determining a differential formulation which approximates said differential equation by substituting the expressed derivative for each of the one or more derivatives of the differential equation;

combining said determined approximate solutions and said differential formulation to obtain a set of algebraic equations;

establishing a characteristic comprising heat conduction of the object using the set of algebraic equations in order to resolve the engineering problem; and

displaying on a computer display a report describing the heat conduction of the object that resolves the engineering problem.

2. The computer-readable medium according to claim 1 , the method:

including at least one known condition which relates to said region and which corresponds to an unknown in said set of algebraic equations;

including the step of substituting each said known condition into said set of algebraic equations, and

including the step of solving said algebraic equations for a further unknown.

3. The computer-readable medium according to claim 2 , wherein each said known condition is a respective boundary condition.

4. The computer-readable medium according to claim 2 , the method including the step of generating a report which presents information generated during said solving step.

5. The computer-readable medium according to claim 2 ,

wherein the physical characteristic is associated with a design property of the object,

wherein the differential equation is representative of the design property of the object, and

including the step of using information generated during said solving step to modify said design property of said object.

6. The computer-readable medium according to claim 1 , the method:

including the step of defining a plurality of nodes in each of said finite elements;

wherein said step of determining said approximate solutions includes the step of determining each of said approximate solutions as a function of a subset of said nodes which are within the finite element corresponding to that approximate solution; and

wherein said step of determining said differential formulation includes the step of determining said differential formulation as a function of a set of node variables representative of the nodes within any of said finite elements.

7. The computer-readable medium according to claim 6 , the method including the step of selecting the number of said node variables so that said differential formulation is a mixed element formulation.

8. The computer-readable medium according to claim 1 , wherein the expansion series is an expansion of Taylor's series.

9. The computer-readable medium according to claim 8 , the method including the step of selecting as said expansion of Taylor's series a first order expansion of Taylor's series.

10. The computer-readable medium according to claim 9 , the method including the step of selecting as said expansion of Taylor's series a first term of said first order expansion of Taylor's series.

11. The computer-readable medium according to claim 1 , wherein said differential formulation is carried out using finite differences with a forward difference approach.

12. The computer-readable medium according to claim 1 , wherein said differential formulation is carried out using finite differences with a backward difference approach.

13. The computer-readable medium according to claim 1 , wherein said differential formulation is carried out using finite differences with a central difference approach.

14. The computer-readable medium according to claim 1 , wherein each said approximate solution is at least a second order solution.

15. The computer-readable medium according to claim 14 , wherein at least one of said approximate solutions is a higher order solution than another of said approximate solutions.

16. A computer-readable medium encoded with a computer program, said computer program being operable when executed to:

receive input comprising a description of a thermally conductive physical object and a differential equation representing heat conduction of the physical object;

divide a region into a plurality of finite elements to resolve an engineering problem, the region representing at least a portion of the physical object analyzed by the engineering problem;

determine for each of said finite elements a respective approximate solution for the differential equation which relates to said region, the approximate solution determined according to piecewise approximation of said region, the differential equation describing a physical property of the object and comprising one or more derivatives;

express each of the one or more derivatives according to a series expansion of each of the one or more derivatives to yield an expressed derivative;

determine a differential formulation which approximates said differential equation by substituting the expressed derivative for each of the one or more derivatives of the differential equation;

combine said determined approximate solutions and said differential formulation to obtain a set of algebraic equations;

establish a characteristic comprising heat conduction of the object using the set of algebraic equations in order to resolve the engineering problem; and

provide display on a computer display a report the characteristic describing the heat conduction of the object that resolves the engineering problem.

17. A computer-readable medium according to claim 16 ,

including at least one known condition which relates to said region and which corresponds to an unknown in said set of algebraic equations; and

wherein said computer program is further operable when executed to substitute each said known condition into said set of algebraic equations, and to solve said algebraic equations for a further unknown.

18. A computer-readable medium according to claim 16 , wherein said computer program is further operable when executed to:

define a plurality of nodes in each of said finite elements;

effect said determination of said approximate solutions in a manner which includes determining each of said approximate solutions as a function of a subset of said nodes which are within the finite element corresponding to that approximate solution; and

effect said determination of said differential formulation in a manner which includes determining said differential formulation as a function of a set of node variables representative of the nodes within any of said finite elements.

19. A computer-readable medium according to claim 16 , wherein said computer program is further operable when executed to effect said determination of said differential formulation in a manner which includes approximating said differential equation based on an expansion of Taylor's series.

20. A system comprising:

a system unit configured to:

receive input comprising a description of a thermally conductive physical object and a differential equation representing heat conduction of the physical object;

divide a region of interest into a plurality of higher order finite elements to resolve an engineering problem, the region representing at least a portion of the physical object analyzed by the engineering problem;

determine for each of said finite elements a respective approximate solution for the differential equation based on piecewise approximation relating to said region, the differential equation describing a physical property of the object and comprising one or more derivatives;

express each of the one or more derivatives according to a series expansion of each of the one or more derivatives to yield an expressed derivative;

determine a differential formulation which approximates a governing equation by substituting the expressed derivative for each of the one or more derivatives of the differential equation;

combine said determined approximate solutions and said differential formulation using mixed order approximation to obtain a set of algebraic equations; and

establish a characteristic comprising heat conduction of the object using the set of algebraic equations in order to resolve the engineering problem; and

a display configured to;

display a report of the characteristic describing the heat conduction of the object that resolves the engineering problem.