STRUCTURAL WEAK SPOT ANALYSIS

• US 20150278414A1
• Filed: 10/23/2013
• Published: 10/01/2015
• Est. Priority Date: 10/24/2012
• Status: Active Grant

• Current Assignee: New York University
• Original Assignee: New York University
• Inventors: Zorin, Denis, Zhou, Qingnan, Panetta, Julian
• Granted Patent: US 10,642,945 B2
• Time in Patent Office: Days
• US Class Current: 1/1
• CPC Class Codes: G06F 30/23 using finite element methods [FEM] or finite difference methods [FDM]

Abstract

A system and methods for weak spot analysis. A mesh shape is preprocessed to approximate an input model for an object. The mesh shape is analyzed with modal analysis to identify weak regions. A method and system for determining weak spots in an object. The method and system uses an optimization problem which is solved to determine a pressure distribution on the object maximizing maximal principal stress by solving a set of optimization problems maximizing stress for each point of the object.

Claims

1. A method of determining mechanically weak regions in an object, comprising the steps of:

preprocessing a mesh shape to approximate an input model for an object;

performing modal analysis on the mesh shape by computing natural vibrational frequencies and modal displacement of the mesh shape; and

using the modal analysis to identify weak regions of the mesh shape.

2. The method as defined in claim 1 further including the step of determining load distribution given the weak regions of the mesh shape.

3. The method as defined in claim 2 wherein the step of determining load distributions is accomplished by the step of using a method of convex optimization.

4. The method as defined in claim 2 wherein the step of determining load distributions is accomplished by the step of using a method of linear programming.

5. The method as defined in claim 2 wherein the step of determining load distribution comprises using a method of nonconvex optimization.

6. The method as defined in claim 1 wherein the method is applied to 3D printing of objects.

7. The method as defined in claim 1 further including the step of providing input data selected from the group consisting of providing shape of the input model, total maximal surface force applied to the object and allowed stress range for object.

8. The method as defined in claim 2 wherein the step of determining load distributions includes determining ways to distribute the load on a surface of the object.

9. The method as defined in claim 2 wherein the step of determining load distribution includes given a maximal stress, finding the load distribution for a minimal total force applied.

10. The method as defined in claim 1 further including the step of optimizing stress distribution knowing the weak regions and total applied force to the object.

11. The method as defined in claim 1 further including the step of separately identifying weak spots from worst case force loads.

12. The method as defined in claim 1 further including the step of applying a global application of the steps to identify multiple ones of the weak regions simultaneously.

13. The method as defined in claim 1 further including the step of using a constrained surface load model for determining maximal stresses of each of the weak regions.

14. The method as defined in claim 1 wherein the step of identifying the weak region using a geometric model including computing local thickness of the object.

15. The method as defined in claim 1 wherein the step of using the model analysis comprises using an elasticity discretization process.

16. The method as defined in claim 1 further including the step of determining worst case force effects via a constrained force analysis performed by applying forces with a positive component normal to a surface of the object with maximal magnitude and constrained total load, thereby enabling modeling of typical force behavior in handling and transporting 3D printed ones of the object.

17. A method of determining weak spots in an object, comprising the steps of:

determining maximal principal stress on each selected point of the object and

obtaining a corresponding pressure distribution and selecting the pressure distribution corresponding to the point for which maximal stress is the greatest thereby determining the weak spots on the object.

18. The method as defined in claim 17 wherein an objective function is further provided to determine optimization, wherein the objective function comprises,

19. The method as defined in claim 18 wherein the maximal principal stress is replaced by a displacement norm ≡

mμ

| in the objective function, thereby optimizing the objective function for maximal displacements of the object.

20. The method as defined in claim 17 wherein a portion of the object is selected for determination of the objective function, thereby establishing maximum stress for a limited area of the object.

21. The method as defined in claim 20 wherein the step of selection comprises performing a computational modal analysis.

22. The method as defined in claim 21 wherein the computational modal analysis comprises computing eigenvectors, μ

, and eigenvalues λ

_i of L wherein,

Lu_i=λ

_iU_i, i=1,2,3 . . .

23. The method as defined in claim 18 wherein the objective function is replaced with an approximate convex function by assuming a primary principal stress as being maximized and other principal stresses are presumed small relative to the primary principal stress.

24. The method as defined in claim 23 wherein solving the convex function is separated into steps of solving for maximal compressive stress and maximal tensile stress.

25. The method as defined in claim 22 wherein stress in the weak region is determined by using a weighted average L2 of all stress values over each of the weak regions, wherein L2 is calculated from the eigenvectors wherein for each of the eigenvectors i, i=1 . . . Mm and each of the weak regions Dij, j−

1 . . . Mn and solving

∫

ω

trσ

dV→

min;

Lu=0 on Ω

,C;

(∇

u+∇

u^T)_n=pn on ∂

Ω

∫

<sub>∂

Ω</sub>

pndA=0,∫

<sub>∂

Ω</sub>

pn×

(x−

x_c)dA=0,

p≧

0,p≦

p_max on ∂

Ω

;

∫

<sub>∂

Ω</sub>

pdA=F_tot

26. The method as defined in claim 17 wherein the objective function is solved by a step of discretization using piecewise-linear finite elements and optimizing,

W^TMBu

where B is a matrix discretizing a strain operator, whereby applying an FEM discretization method to the vector of displacements u, yields a vector of values of strain, with 6 components per strain value, and M is a matrix representing the application of a rank-4 tensor C to strain values, thereby yielding a vector of stress components.

27. The method as defined in claim 26 including the step of eliminating displacements u, thereby accelerating determination of object pressures, p.

28. A non-transitory computer-accessible medium having stored thereon computer-executable instructions for determining weak spots in an object, wherein, when a computer hardware arrangement executes the instructions, the computer arrangement is configured to perform procedures comprising:

receiving information related to an objective function for processing to determine a maximal principal stress on the object; and

receiving information related to solving the objective function by determining a solution for each selected point on the object by identifying minimal stress for the objective function at the selected point

29. The non-transitory computer-accessible medium as defined in claim 28 wherein receiving information related to determining optimization via an objective function, wherein the objective function comprises,

30. The non-transitory computer-accessible medium as defined in claim 29 wherein the maximal principal stress is replaced by a displacement norm |mμ

| in the objective function, thereby optimizing the objective function for maximal displacements of the object.

31. The non-transitory computer-accessible medium as defined in claim 28 further comprising receiving information related to selecting a portion of the object for determination of the objective function, thereby establishing maximum stress for a limited area of the object.

32. The non-transitory computer-accessible medium as defined in claim 31 wherein receiving information related to selecting a portion comprises receiving information related to performing a computational modal analysis.

33. The non-transitory computer-accessible medium as defined in claim 32 wherein the computational modal analysis comprises computing eigenvectors, μ

, and eigenvalues λ

_i of L wherein,

Lu_i=λ

_iU_i, i=1,2,3 . . .

43. The method as defined in claim 33 wherein the objective function is solved by a step of discretization using piecewise-linear finite elements and optimizing:

w^TMBu

where B is a matrix discretizing a strain operator, whereby applying an FEM discretization method to the vector of displacements u, yields a vector of values of strain, with 6 components per strain value, and M is a matrix representing the application of a rank-4 tensor C to strain values, thereby yielding a vector of stress components.

34. A method of determining weak spots in an object, comprising the steps of:

providing constrained optimization problem with maximum principal stress as the objective function and with pressure distribution;

solving for the pressure distribution by determining a solution maximizing stress at each point on the object; and

solving the nonlinear optimization problem by sampling admissible pressure distributions used to initialize the nonlinear solver.

35. The method as defined in claim 34 wherein the optimization problem comprises,

36. The method as defined in claim 35 wherein the maximal principal stress is replaced by a displacement norm |mμ

| in the objective function, thereby optimizing the objective function for maximal displacements of the object.

37. The method as defined in claim 34 wherein a portion of the object is selected for determination of the objective function, thereby establishing maximum stress for a limited area of the object.

38. The method as defined in claim 37 wherein the step of selection comprises performing a computational modal analysis.

39. The method as defined in claim 38 wherein the computational modal analysis comprises computing eigenvectors, and eigenvalues λ

_i of L wherein,

Lu_i=λ

_iU_i, i=1,2,3 . . .

40. The method as defined in claim 35 wherein the objective function is replaced with an approximate convex function by assuming the sum of primary principal stresses as being maximized and other principal stresses are presumed small relative to the largest principal stress.

41. The method as defined in claim 40 wherein solving the convex function is separated into steps of solving for maximal compressive stress and maximal tensile stress.

42. The method as defined in claim 39 wherein average stress in the weak region is determined by using a weighted average of all stress values over each of the weak regions with weight equal to L2 norm of the stress corresponding to eigenvectors wherein for each of the eigenvectors i, i=1 . . . Mm and each of the weak regions Dij, j−

1 . . . Mn and solving

∫

ω

trσ

dV→

min;

Lu=0 on Ω

,C;

(∇

u+∇

u^T)n=pn on ∂

Ω

∫

<sub>∂

Ω</sub>

pndA=0,∫

<sub>∂

Ω</sub>

pn×

(x−

x_c)dA=0,

p≧

0,p≦

p_max on ∂

Ω

;

∫

<sub>∂

Ω</sub>

pdA=F_tot

44. The method as defined in claim 35 including the step of eliminating displacements u, thereby accelerating determination of object pressures, p.