Topology optimization process
Solving large models

Large FEA models, containing many millions of finite elements, are frequently encountered in practical topology optimization. Since engineering optimization is always a cyclic process, large-scale models may represent a serious obstacle preventing to get a good result within reasonable time. In this respect ProTOp offers three technologies that can be combined to solve large-scale problems very efficiently. These three technologies are

• the semi-active element technology,
• the interactive element removal functionality, and
• the interactive localized mesh refinement functionality.

Note that the semi-active element technology is always engaged automatically without user intervention. On the other hand, the element removal and mesh refinement has to be requested by the user when desired.

The fundamental idea to assure efficient optimization is to start the optimization process with a FE mesh of low density. As the optimization progresses, material is typically removed, resulting in a reduced volume of the optimized part. At this stage the semi-active element technology already takes care of boosting the performance by exploiting the partially void design. As the volume of the structure is further reduced, the user can engage the element removal tool to ged rid of finite elements that are not needed any more. Finally, as the current design tends to get its final topology, the user can engage the localized mesh refinement tool in order to get a fine FE mesh that will deliver a good and reliable result.

An outline of the recommended procedure is given in the following.

Preparing an adequate low-density mesh

To address large-scale problems and get the most out of ProTOp, one should start with a FEA model containing a mesh of low to moderate density. When preparing this mesh, care should be taken that the elements are still fine enough to:

• describe the boundary surfaces geometry with enough accuracy (it should be noted that subsequent mesh refinements can not improve the geometrical representation of boundary surfaces), and
• enable acceptable topology optimization results in the initial stage of topology optimization (mesh roughness should not influence too heavily the generated topology of the structure).

Let us consider a bracket structure as an illustrative example. To start the optimization process, a low-density mesh is prepared exhibiting the following properties:

• Elements: 0.57 million
• DOF: 0.32 million

Figure. The initial low-density mesh; approximately 0.57 million elements and 0.32 million DOFs

Optimizing the low-density-mesh model

The optimization should be started by using the low-density mesh, which in this example contains less than 1 million tetrahedral finite elements. So, the optimization problem can be solved rather quickly, and the obtained result (for 35% volume part) is shown below.

Figure. Optimization result obtained by the low-density mesh (35% volume part)

For illustration it might be worth to list the approximate performance data (desktop PC, i7 CPU, 4 cores):

• Max RAM consumption: 1.5 GB
• Total run time (4 load cases, 40 cycles): 00:04:07

Stripping and refining the mesh

After removing some part of the material, it is usually a good idea to stop the optimizer and do some work on the mesh. Specifically, there are two operations that can be quite useful, as follows:

• Mesh stripping. After removing a certain amount of material there might be some void regions for which we can be sure that will never be filled with material again. In such a case the mesh should be stripped in order to reduce the number of involved finite elements. Note that this step also reduces the domain of the optimized structure.
• Mesh refinement. Regardless of the optional mesh stripping, the initially used low-density mesh should always be refined for the final stage of optimization. This refinement must ensure that the cut surfaces (and their neighborhood) of the optimal topology design will be meshed with reasonably fine finite elements.

Note that mesh stripping is not obligatory; it just reduces the model size and consequently speeds up all subsequent procedures. Mesh refinement, on the other hand, is typically absolutely necessary to assure good and reliable results.

In this example the refined mesh was obtained localized mesh refinement along the material-void boundaries. The refined mesh exhibits the following properties:

• Elements: 2.68 million
• DOF: 1.40 million

Optimizing the refined high-density-mesh model

Because the continued optimization process will run from current design (with a lot of void elements), the semi-active technology will assure that a lot of finite elements will never get activated at all. Therefore, the consumed RAM and required CPU time will remain far below the levels that would actually be needed when starting the optimization from a full-material design. This means that the optimization process will proceed very efficiently until the final design (figure below) is obtained.

In our example, continuing the optimization process with the refined mesh and running a few cycles would result in the final design as shown in the figure below.

Figure. Optimization result obtained after continuing optimization with the refined high-density mesh (35% volume part)

For illustration it might be worth to list the approximate performance data for this second stage of optimization with the refined mesh (desktop PC, i7 CPU, 4 cores):

• Max RAM consumption: 4.0 GB
• Total run time (4 load cases, 14 cycles): 00:07:39

NOTE. By using the proposed technique, the optimal design of a nearly 3-million-elements model has been obtained by utilizing only 4.0 GB of RAM and within less than 12 minutes. If the optimization would be started with the dense mesh from full-material design, the process would require about 12 GB of RAM and dramatically more CPU time.

Alternative options

In some cases it may be desired that the boundary surfaces of the optimized model are described by geometrical accuracy being higher than the one delivered by the low-density mesh. The figures below illustrate a detail of two meshes, generated by the mesher from the geometrical CAD model.

Figure. Low-density mesh detail - geometrical accuracy of boundary surfaces is relatively low

Figure. High-density mesh detail - geometrical accuracy of boundary surfaces is acceptable

One can see that the boundary surface geometry of the low-density mesh is of a relatively low quality which might not be acceptable for the final design. In such cases it is still recommended to start optimization from the low-density mesh, but now we have to follow a somewhat different procedure. The recommended steps can be outlined as follows:

• 1. Prepare two FEA meshes. Run the mesher twice to generate two different meshes from your CAD model: a low-density and a high-density mesh.
• 2. Optimize the low-density-mesh model. This will generate quickly some rough approximation of the optimal design.
• 3. Create a new session and import the high-density mesh. The new session should be created as a copy of the currently active session that holds the low-density-mesh result.
• 4. Map the low-density-mesh result into the newly created session. This will import the approximate optimal design and make the continued optimization process very efficient.
• 5. Proceed with optimization as usual.

NOTE. By using the proposed technique, a high-density-mesh model with geometrically accurate boundary surfaces may be optimized in a very efficient way without the need to ever load the high-density full-material model into the FEA solver.