Simulation-Free Reduction Basis Interpolation to Reduce Parametrized Dynamic Models of Geometrically Non-linear Structures

Abstract

Virtual design studies for the dynamics of structures that undergo large deformations, such as wind turbine blades or Micro-Electro-Mechanical Systems (MEMS), can be a tedious task. Such studies are usually done with finite element simulations. The equations of motion that result from the finite element discretization typically are high-dimensional and nonlinear. This leads to high computation costs because the high-dimensional nonlinear stiffness term and its Jacobian must be evaluated at each Newton-Raphson iteration during time integration. Model reduction can overcome this burden by reducing the high-dimensional model to a smaller problem. This is done in two steps: First, a Galerkin projection on a reduction basis, and, second, hyperreduction of the geometric nonlinear restoring force term.

The first step, namely finding a proper reduction basis, can be performed by either simulation-based or simulation-free methods. While simulation-based methods, such as the Proper Orthogonal Decomposition (POD), rely on costly preliminary simulations of full high-dimensional models, simulation-free methods are much cheaper in computation. For this reason, simulation-free methods are more desirable for design studies where the amount of the so called ‘offline costs’ for reduction of the high-dimensional model are of high interest. However, simulation-free reduction bases are dependent on the system’s properties, and thus depend on design parameters that typically change for each design iteration. This dependence must be taken into account if the parameter space of interest is large.

This contribution shows how design iterations can be performed without the need for expensive simulations of the high-dimensional model. We propose to sample the parameter space, compute simulation-free reduction bases at the sample points and interpolate the bases at new parameter points. As hyperreduction technique, the Energy Conserving Sampling and Weighting method and the Polynomial expansion are used for hyperreduction of the nonlinear term. In this step, we also avoid simulations of the high-dimensional nonlinear model. The coefficients of the hyperreduction are updated in each design iteration for the new reduction bases.

A simple case study of a shape parameterized beam shows the performance of the proposed method. The case study also accounts for a last challenge that occurs in models that are parametric in shape: The topology of the finite element mesh must be maintained during the design iterations. We face this challenge by using mesh morphing techniques.