A very common problem to solve in light weight optimization is to design the part for maximum stiffness subjected to a certain fraction of material cost saving. It is a topology-based optimization scheme in which few elements are iteratively removed from the model in such a way that stiffness of remaining structure retains its maximum possible stiffness. Sounds Simple!!! The problem is easy to set up and visualize. However, the underlying iterative scheme has its own story, and we will take a deep dive in it. Please note that we are addressing this problem in our Flagship FEA based optimization solution called TOSCA.
The objective of this problem is global stiffness for which the correct design response is “strain energy”. The design response quantifies the global stiffness of the structure. However, its correlation with stiffness is not trivial. Many other factors determine how to correctly use the strain energy response to maximize the stiffness of the structure. We will demonstrate it through a simple problem.
PROBLEM DEFINITION:
Consider a simple plane stress problem of a plate clamped on left edge and the lower right vertex is subjected to a load which could be a force or an enforced displacement. The objective is to maximize the stiffness of structure after 55 percent material removal. How to use the “strain energy” correctly to set up this problem? We need to consider all the possible cases.
First, let’s talk briefly about sensitivities. Optimization can be done either using Abaqus or Tosca sensitivities. If Abaqus CAE is used for modeling, Abaqus sensitivities are checked by default as below. If Tosca GUI is used instead, Tosca sensitivities are checked by default. Either way, it’s a user choice.
Now lets consider the two cases:
Case1: Enforced displacement of 4 units at the location shown above with Abaqus sensitivities.
Case2: Point load of 40 units of 4 at the location shown above with Abaqus sensitivities.
OBSERVATIONS:
Case1: Maximum stiffness is achieved by Minimizing Strain Energy. The objective function history and results look as below.
Case2: Maximum stiffness is achieved by Minimizing Strain Energy. The objective function history and results look as below.
A critical observation is that the objective function changes its sign from positive in first case to negative in second case though the problem is physically identical. However, Tosca treats the two scenarios as different, only from a numerical perspective.
Another, somewhat less popular response to maximize stiffness is the “energy stiffness measure”. It is recommended for prescribed displacement loads but it does work for force-based loading as well. The difference between the two response types is insignificant but worth noting.
Strain Energy is defined as ½*P*u where P is external loading and u is deflection
Stiffness Energy measure is defined as ½*R*U where R is reaction force and U is enforced displacement.
In case of combined direct and enforced displacement loadings, the response function is calculated as follows:
Strain Energy Response = ½*P*u+1/2*R*U
Stiffness Energy Response = ½*P*u-1/2*R*U
That explains why objective function changes sign if load type is flipped from one to another in case of a strain energy response function. The stiffness-energy response will not flip in sign because the two strain energy terms are subtracted and reaction force is always opposite to the applied load.
Tosca UI offers yet another function for stiffness called “compliance”. It is directly proportional to strain energy in load-based problems and inversely proportional to strain energy in enforced displacement-based problems.
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