Contenido

Existen varias metodologías de implementación que se han utilizado para resolver problemas de optimización topológica.

Solve with discrete/binary variables

The solution of topology optimization problems in a discrete sense is done by discretizing the design domain in finite elements. The material densities within these elements are then treated as the variables of the problem. In this case, a material density of 1 indicates the presence of material, while 0 indicates its absence. Because the achievable topological complexity of the design depends on the number of elements, a large number is preferred. A large number of finite elements increases the achievable topological complexity, but comes at a cost. First, the finite element system solution becomes more expensive. Second, there are no algorithms that can handle a large number (several thousand elements is not uncommon) of discrete variables with multiple constraints. Furthermore, they are not very sensitive to parameter variations.[1]​ In the literature, problems with up to 30,000 variables have been reported.[2]​.

Solve the problem with continuous variables

The above-mentioned complexities of solving topology optimization problems using binary variables have led the community to look for other options. One of them is the modeling of densities with continuous variables. Material densities can now also reach values ​​between 0 and 1. There are gradient-based algorithms that handle large amounts of continuous variables and multiple constraints. However, material properties must be modeled in a continuous environment. This is done through interpolation. One of the most implemented interpolation methodologies is the SIMP method (from English for Solid Isotropic Material with Penalty).[3]​ This interpolation is a power law: . Interpolates the Young's modulus of the material to the scalar selection field. The value of the penalty parameter is taken to be , generally. This has been shown to confirm the microstructure of the materials.[4]​ In the SIMP method, a lower bound is added to the Young's modulus, , to ensure that the derivatives of the objective function are non-zero when the density becomes zero. The larger the penalty factor, the larger the SIMP algorithm's penalty when using non-binary densities. Unfortunately, the penalty parameter also introduces non-convexities to the problem.

Commercial software

There are several commercial topology optimization software on the market. Most use topology optimization as a guide to optimal design and require manual reconstruction of the geometry. There are some solutions that generate optimal designs ready for additive manufacturing.[5]​.

Structural flexibility

A rigid structure is one that presents the least possible displacement under certain boundary conditions. A global measure of displacements is the strain energy (also called flexibility) of the structure under the prescribed boundary conditions. The lower the strain energy, the higher the stiffness of the structure. Therefore, the objective function of the problem is to minimize the strain energy.

In general, it can be seen that the greater the amount of material, the less the deflection will be, since there will be more material to resist the loads. Therefore, optimization requires an opposite constraint: the volume constraint. This is actually a cost factor, since you don't want to invest a lot of money in the material. To obtain the total material used, an integration of the selection field over the volume can be performed.

Finally, the differential equations that govern elasticity are introduced to obtain the final formulation of the problem:

subject to:.

However, a simple implementation in the finite element framework of such a problem is not yet feasible due to issues such as:

1. • Mesh dependency: The design obtained in one mesh can be very different from that obtained in another. Design features become more complex as the mesh is refined.[6]​.

2. • Numerical instabilities: a small change in an input parameter can produce a large change in the calculated solution.[7]​.

Some techniques, such as filtering, "Core (Digital Image Processing)") based image processing,[8]​ are currently used to mitigate some of these problems. Although for a long time this appeared to be a purely heuristic approach, theoretical connections with non-local elasticity have been made to support the physicality of these methods.[9]​.

Multiphysics problems

Fluid-structure interaction") is a strongly coupled phenomenon that refers to the interaction between a stationary or moving fluid and an elastic structure. Many engineering applications and natural phenomena are subject to fluid-structure interactions, so considering these effects is crucial in the design of various engineering applications. Topology optimization for fluid-structure interaction problems has been studied in, for example, references,[10]​[11]​ [12]​ and.[13]​ Below are design solutions. for different Reynolds numbers. The design solutions depend on the flow, indicating that the coupling between the fluid and the structure is solved in the design problems.

Thermoelectricity is a multiphysics phenomenon that refers to the interaction and coupling between electrical and thermal energy in semiconductor materials. Thermoelectric energy conversion can be described by two independent effects: the Seebeck effect and the Peltier effect. The Seebeck effect refers to the conversion of thermal energy into electrical energy, and the Peltier effect refers to the conversion of electrical energy into thermal energy. By spatially distributing two thermoelectric materials in a two-dimensional design space using a topology optimization methodology,[14] it is possible to surpass the performance of constituent thermoelectric materials for refrigerators and thermoelectric generators.[15]

3F3D: Form follows force in 3D printing

The current proliferation of 3D printing technology has allowed designers and engineers to use topology optimization techniques when designing new products. Topology optimization, combined with 3D printing, can result in lower weight, improved structural performance and a reduction in the design-to-manufacturing cycle, as designs, while efficient, may not be viable with more traditional manufacturing techniques.

Internal contact

Internal contact can be included in topology optimization by applying the third-medium contact method.[17]​ [18]​ The third-medium contact (TMC) method is an implicit, continuous, and differentiable contact formulation. This makes TMC suitable for use with gradient-based topology optimization approaches. Both monolithic[19]​ and staggered approaches,[16][20]​ most common in topology optimization, have been used to create various designs with internal contact. Recently, thermal contact has been included in the TMC topology optimization framework [21].

• - Topology optimization animations.

Contenido

Existen varias metodologías de implementación que se han utilizado para resolver problemas de optimización topológica.

Solve with discrete/binary variables

The solution of topology optimization problems in a discrete sense is done by discretizing the design domain in finite elements. The material densities within these elements are then treated as the variables of the problem. In this case, a material density of 1 indicates the presence of material, while 0 indicates its absence. Because the achievable topological complexity of the design depends on the number of elements, a large number is preferred. A large number of finite elements increases the achievable topological complexity, but comes at a cost. First, the finite element system solution becomes more expensive. Second, there are no algorithms that can handle a large number (several thousand elements is not uncommon) of discrete variables with multiple constraints. Furthermore, they are not very sensitive to parameter variations.[1]​ In the literature, problems with up to 30,000 variables have been reported.[2]​.

Solve the problem with continuous variables

The above-mentioned complexities of solving topology optimization problems using binary variables have led the community to look for other options. One of them is the modeling of densities with continuous variables. Material densities can now also reach values ​​between 0 and 1. There are gradient-based algorithms that handle large amounts of continuous variables and multiple constraints. However, material properties must be modeled in a continuous environment. This is done through interpolation. One of the most implemented interpolation methodologies is the SIMP method (from English for Solid Isotropic Material with Penalty).[3]​ This interpolation is a power law: . Interpolates the Young's modulus of the material to the scalar selection field. The value of the penalty parameter is taken to be , generally. This has been shown to confirm the microstructure of the materials.[4]​ In the SIMP method, a lower bound is added to the Young's modulus, , to ensure that the derivatives of the objective function are non-zero when the density becomes zero. The larger the penalty factor, the larger the SIMP algorithm's penalty when using non-binary densities. Unfortunately, the penalty parameter also introduces non-convexities to the problem.

Commercial software

There are several commercial topology optimization software on the market. Most use topology optimization as a guide to optimal design and require manual reconstruction of the geometry. There are some solutions that generate optimal designs ready for additive manufacturing.[5]​.

Structural flexibility

A rigid structure is one that presents the least possible displacement under certain boundary conditions. A global measure of displacements is the strain energy (also called flexibility) of the structure under the prescribed boundary conditions. The lower the strain energy, the higher the stiffness of the structure. Therefore, the objective function of the problem is to minimize the strain energy.

In general, it can be seen that the greater the amount of material, the less the deflection will be, since there will be more material to resist the loads. Therefore, optimization requires an opposite constraint: the volume constraint. This is actually a cost factor, since you don't want to invest a lot of money in the material. To obtain the total material used, an integration of the selection field over the volume can be performed.

Finally, the differential equations that govern elasticity are introduced to obtain the final formulation of the problem:

subject to:.

However, a simple implementation in the finite element framework of such a problem is not yet feasible due to issues such as:

1. • Mesh dependency: The design obtained in one mesh can be very different from that obtained in another. Design features become more complex as the mesh is refined.[6]​.

2. • Numerical instabilities: a small change in an input parameter can produce a large change in the calculated solution.[7]​.

Some techniques, such as filtering, "Core (Digital Image Processing)") based image processing,[8]​ are currently used to mitigate some of these problems. Although for a long time this appeared to be a purely heuristic approach, theoretical connections with non-local elasticity have been made to support the physicality of these methods.[9]​.

Multiphysics problems

Fluid-structure interaction") is a strongly coupled phenomenon that refers to the interaction between a stationary or moving fluid and an elastic structure. Many engineering applications and natural phenomena are subject to fluid-structure interactions, so considering these effects is crucial in the design of various engineering applications. Topology optimization for fluid-structure interaction problems has been studied in, for example, references,[10]​[11]​ [12]​ and.[13]​ Below are design solutions. for different Reynolds numbers. The design solutions depend on the flow, indicating that the coupling between the fluid and the structure is solved in the design problems.

Thermoelectricity is a multiphysics phenomenon that refers to the interaction and coupling between electrical and thermal energy in semiconductor materials. Thermoelectric energy conversion can be described by two independent effects: the Seebeck effect and the Peltier effect. The Seebeck effect refers to the conversion of thermal energy into electrical energy, and the Peltier effect refers to the conversion of electrical energy into thermal energy. By spatially distributing two thermoelectric materials in a two-dimensional design space using a topology optimization methodology,[14] it is possible to surpass the performance of constituent thermoelectric materials for refrigerators and thermoelectric generators.[15]

3F3D: Form follows force in 3D printing

The current proliferation of 3D printing technology has allowed designers and engineers to use topology optimization techniques when designing new products. Topology optimization, combined with 3D printing, can result in lower weight, improved structural performance and a reduction in the design-to-manufacturing cycle, as designs, while efficient, may not be viable with more traditional manufacturing techniques.

Internal contact

Internal contact can be included in topology optimization by applying the third-medium contact method.[17]​ [18]​ The third-medium contact (TMC) method is an implicit, continuous, and differentiable contact formulation. This makes TMC suitable for use with gradient-based topology optimization approaches. Both monolithic[19]​ and staggered approaches,[16][20]​ most common in topology optimization, have been used to create various designs with internal contact. Recently, thermal contact has been included in the TMC topology optimization framework [21].

• - Topology optimization animations.