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:

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].

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:

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].