Component based model order reduction with mortar tied contact for nonlinear quasi-static mechanical problems

Abstract

In this work, we present a model order reduction technique for nonlinear structures assembled from components.The reduced order model is constructed by reducing the substructures with proper orthogonal decomposition and connecting them by a mortar-tied contact formulation. The snapshots for the substructure projection matrices are computed on the substructure level by the proper orthogonal decomposition (POD) method. The snapshots are computed using a random sampling procedure based on a parametrization of boundary conditions. To reduce the computational effort of the snapshot computation full-order simulations of the substructures are only computed when the error of the reduced solution is above a threshold. In numerical examples, we show the accuracy and efficiency of the method for nonlinear problems involving material and geometric nonlinearity as well as non-matching meshes. We are able to predict solutions of systems that we did not compute in our snapshots.

Figures (24)

• Figure 1: Illustration of a domain, that is composed of two subdomains $\Omega_0^1$ and $\Omega_0^2$, with the Neumann boundaries $\Gamma_\sigma^1$, and $\Gamma_\sigma^2$ and Dirichlet boundaries $\Gamma_u^1$, and$\Gamma_u^2$. At the contact interface $\Gamma_c^{1,2}$, the two tied contact conditions are displayed.
• Figure 2: Geometry, mesh and boundary conditions of the $2 \times 3$ example. The substructure on the bottom right has a finer mesh than the other substructures.
• Figure 3: Geometry, mesh and boundary conditions of the $3 \times 3$ example. The blue substructures have a Young's modulus of $E=80 \: \rm GPa$, the gray substructures of $E=20 \: \rm GPa$.
• Figure 4: Illustration of the snapshot parametrization.
• Figure 5: Decay of the normalized singular values (left), and decay of the mean error of ten unseen test samples over the number of modes $m$. The blue area is the difference between the minimum and maximum mean displacement error.