Systems and methods for finite element based topology optimization

1. A method of providing an optimal minimum mass topology for a structure based on a set of design criteria including at least one support point and at least one force to be applied to the structure, said method comprising executing software instructions to perform the steps of:

identifying a plurality of nodes within a structure design domain, and assigning an initial density value to said plurality of nodes;

conducting a finite element analysis on said nodes;

determining one of a stress intensity or strain energy values for each node;

ranking the nodes by relative stress intensity or strain energy values;

adjusting the density value for each node; and

repeating the steps of conducting a finite element analysis on said nodes, wherein the step of adjusting each density value for each node is performed according to a family of statistical distribution functions that gradually transition to a bimodal distribution wherein nodes are either fully dense or void thereby providing an optimal topology.

2. The method as claimed in claim 1 , wherein at each intermediate iteration, the relative nodal densities are assigned by ranking one of the stress intensity or strain energy values at each node and adjusting the relative nodal densities such that relative density is increased in regions with high stress intensity or strain energy, and is decreased in regions of low stress intensity or strain energy with the overall density distribution assigned according to the prescribed probability function for that iteration.

3. The method as claimed in claim 1 , wherein the design domain for the optimal minimum mass structure is at least two dimensional, and wherein the finite element model is at least two dimensional.

4. The method as claimed in claim 1 , wherein the optimal minimum mass topology is determined for a structure in which the material strength in tension differs from that in compression, wherein convergence to optimal minimum mass topology is guaranteed by imposing artificial and different material modulus values for regions of tensile and compressive stress states such that the resulting strain field satisfies minimum mass optimality criteria for structures in which the material strength in tension differs from that in compression.

5. The method as claimed in claim 4 , wherein the structure includes a fiber reinforced material.

6. The method as claimed in claim 1 , wherein the optimal minimum mass topology is determined for dual material structures in which the materials comprise different strengths, wherein convergence to optimal minimum mass topology is guaranteed by imposing artificial and different material modulus values for regions of tensile and compressive stress states such that the resulting strain field satisfies minimum mass optimality criteria for dual material structures.

7. The method as claimed in claim 6 , wherein at least one of the material structures includes a fiber reinforced material.

8. The method as claimed in claim 1 , wherein the optimal minimum mass topology for structures subjected to multiple alternative loadings is determined by, at each iteration, performing multiple finite element analyses corresponding to multiple alterative loadings, and ranking the maximum stress intensity or strain energy for each node.

9. A method of providing an optimal minimum mass topology for a structure based on a set of design criteria including at least one support point and at least one force to be applied to the structure, said method comprising executing software instructions to perform the steps of:

conducting a finite element analysis on a plurality of nodes within a structure design domain, each of said plurality of nodes having an initial density value;

determining one of a stress intensity or strain energy value for each node;

ranking the nodes by relative stress intensity or strain energy values; adjusting the density value for each node; and

repeating the steps of conducting a finite element analysis on said nodes, wherein the step of adjusting each density value for each node is performed according to a family of statistical distribution functions that gradually transition to a bimodal distribution wherein nodes are either fully dense or void thereby providing an optimal topology, wherein the family of probability distributions include a constant probability mean equal to ρ o where p represents partial density, wherein the design domain is initially defined by P(ρ=ρ o )=1 and σ 2 =0, wherein P represents the probability that ρ=ρ 0 anywhere in the design domain and σ 2 is the probability variance, wherein after a prescribed number of iterations the probability density distribution becomes bimodal with probability values P(ρ=ρ o )=1−ρ o , P(ρ=1)=ρ o and σ 2 =ρ o (1−ρ o ), and wherein a prescribed number of intermediate continuous probability distributions are used to define the material density distribution at each iteration, wherein average density remains constant, the distribution becomes progressively bimodal and the variance increases monotonically from zero towards ρ o (1−ρ o ).

10. The method as claimed in claim 9 , wherein at each intermediate iteration, the relative nodal densities are assigned by ranking one of the stress intensity or strain energy values at each node and adjusting the relative nodal densities such that relative density is increased in regions with high stress intensity or strain energy, and is decreased in regions of low stress intensity or strain energy with the overall density distribution assigned according to the prescribed probability function for that iteration.

11. The method as claimed in claim 9 , wherein the design domain for the optimal minimum mass structure is at least two dimensional, and wherein the finite element model is at least two dimensional.

12. The method as claimed in claim 9 , wherein the optimal minimum mass topology is determined for a structure in which the material strength in tension differs from that in compression, wherein convergence to optimal minimum mass topology is guaranteed by imposing artificial and different material modulus values for regions of tensile and compressive stress states such that the resulting strain field satisfies minimum mass optimality criteria for structures in which the material strength in tension differs from that in compression.

13. The method as claimed in claim 9 , wherein the optimal minimum mass topology is determined for dual material structures in which the materials comprise different strengths, wherein convergence to optimal minimum mass topology is guaranteed by imposing artificial and different material modulus values for regions of tensile and compressive stress states such that the resulting strain field satisfies minimum mass optimality criteria for dual material structures.

14. A method of providing an optimal minimum mass topology for a structure based on a set of design criteria including at least one support point and at least one force to be applied to the structure, said method comprising executing software instructions to perform the steps of:

identifying a plurality of nodes within a structure design domain, each of said plurality of nodes having an initial density value;

conducting a finite element analysis on said nodes;

determining one of a stress intensity or strain energy value for each node; ranking the nodes by relative stress intensity or strain energy values;

adjusting the density value for each node; and

repeating the steps of conducting a finite element analysis on said nodes, wherein the step of adjusting each density value for each node is performed according to a family of statistical distribution functions that gradually transition to a bimodal distribution wherein nodes are either fully dense or void thereby providing an optimal topology, wherein a particular Beta probability distribution is prescribed at each iteration, and is selected to ensure that the total mass is maintained at a constant value for all finite element iterations, and to provide a smooth, gradual transition from the initial uniform partially dense distribution to the final bimodal relative density distribution that defines the optical topology.

15. The method as claimed in claim 14 , wherein at each intermediate iteration, the relative nodal densities are assigned by ranking one of the stress intensity or strain energy values at each node and adjusting the relative nodal densities such that relative density is increased in regions with high stress intensity or strain energy, and is decreased in regions of low stress intensity or strain energy with the overall density distribution assigned according to the prescribed probability function for that iteration.

16. The method as claimed in claim 14 , wherein the design domain for the optimal minimum mass structure is at least two dimensional, and wherein the finite element model is at least two dimensional.

17. The method as claimed in claim 14 , wherein the optimal minimum mass topology is determined for a structure in which the material strength in tension differs from that in compression, wherein convergence to optimal minimum mass topology is guaranteed by imposing artificial and different material modulus values for regions of tensile and compressive stress states such that the resulting strain field satisfies minimum mass optimality criteria for structures in which the material strength in tension differs from that in compression.

18. The method as claimed in claim 14 , wherein the optimal minimum mass topology is determined for dual material structures in which the materials comprise different strengths, wherein convergence to optimal minimum mass topology is guaranteed by imposing artificial and different material modulus values for regions of tensile and compressive stress states such that the resulting strain field satisfies minimum mass optimality criteria for dual material structures.