Lumped Mass Modeling for Local-Mode-Suppressed Element Connectivity

For successful topology design optimization of crashworthy “continuum” structures, unstable element-free and local vibration mode-free transient analyses should be ensured. Among these two issues, element instability was shown to be overcome if a recently-developed formulation, the element connectivity parameterization (ECP) is employed.

On the way to the ultimate crashworthy structure optimization, we are now developing a local mode-free topology optimization formulation that can be implemented in the ECP method. In fact, the local mode-freeing strategy developed here can be also used directly for the standard element density method.

Local modes are artificial, numerical modes resulting from the intrinsic modeling technique of the topology optimization method. Even with existing local mode controlling techniques, the convergence of the topology optimization of vibrating structures, especially experiencing large structural changes, appears to be still poor.

In ECP, the nodes of the domain-discretizing elements are connected by zero-length one-dimensional elastic links having varying stiffness. For computational efficiency, every elastic link is now assumed to have two lumped masses at its ends. Choosing appropriate penalization functions for lumped mass and link stiffness is important for local mode-free results.

However, unless the objective and constraint functions are carefully selected, it is difficult to obtain clear black-and-white results. It is shown that the present formulation is also successful in design problems involving self-weight.In terms of computation time, the I-ECP method, a newly-developed version of ECP, is much more efficient because the degrees of freedom of the element-connectivity parameterizing links are eliminated in element level before the total system matrix is assembled.

In terms of implementation, however, the E-ECP is easier to use because the sensitivity analysis in E-ECP does not require the explicit expression of the (tangent) stiffness matrix of continuum finite elements. Therefore, any finite element code, including commercial codes, can be readily used for the ECP implementation.

The key ideas and characteristics of these methods will be presented in this paper.