Model Order Reduction using the Discrete Empirical Interpolation Method

Model Order Reduction using the Discrete Empirical Interpolation Method

Dedden, R.J.

Tiso, P. (mentor)

Mechanical, Maritime and Materials Engineering

Precision and Microsystems Engineering

2012-11-07

The invention of the computer opened new research fields in physics and engineering. One of the developments is the Finite Element Method (FEM). It has matured over the last decades, resulting in complex applications with many Degrees Of Freedom (dofs). The large number of dofs made the finite element problems expensive to solve. To make Finite Element Analysis (FEA) less expensive, Model Order Reduction (MOR)-techniques have been developed. These techniques approximate the original full order problem by a system of lower dimension. This is done by writing the displacement field in terms of a set of reduced coordinates. For linear problems, this can considerably reduce the amount of computations required. However, the number of computations for nonlinear problems can not be reduced in the same way. This thesis focusses on a special type of nonlinear problems, namely geometrically nonlinear problems. In general, the internal force of such systems consists of both linear and nonlinear contributions. Whereas the linear contribution can be reduced with the reduced coordinates, the evaluation of the nonlinear term requires the full order displacement field to be known. This forms a bottleneck in the computations in terms of computation time. A solution to the bottleneck of the nonlinear term is found in the Discrete Empirical Interpolation Method (DEIM). The DEIM is used in combination with a Proper Orthogonal Decomposition (POD), yielding the POD-DEIM reduction of the nonlinear part of the internal force. The POD-DEIM reduction approximates the space of the nonlinear part of the internal forces by a POD. The resulting reduced subspace is then interpolated with DEIM. In practice, the POD-DEIM reduction bores down to an approximation of the nonlinear part of the internal force that requires only a few components (dofs) of the nonlinear internal force vector. Using these few components, the remaining components are approximated through interpolation. The advantage of DEIM over other reduction methods that are able to reduce nonlinear terms in systems of equations, is that the DEIM is able to select its components or collocation points from the full set of dofs automatically. This has the advantage that no user input is required in the reduction of the nonlinear finite element model. The DEIM has been tested on several load case examples, using mass-spring systems and geometrically nonlinear bar and beam elements. Two different approaches have been successfully applied to bar elements. The direct approach applies the POD-DEIM reduction directly on the nonlinear term in the system of equations that results from the finite element model. The unassembled approach applies POD-DEIM to the nonlinear terms before the finite element model is assembled. Both methods showed that accurate results can be obtained while using the nonlinear response of only a limited number of dofs. Although the potential of DEIM for geometrically nonlinear finite elements has been shown, some difficulties will have to be overcome before DEIM can be used on general problems. First of all, both the direct and the unassembled approach have some inherent inefficiencies that limit the computational savings. The direct approach generally selects a high number of finite elements, because the selected dofs are often shared by many elements. This problem is solved using the unassembled approach. However, the subspace of unassembled nonlinear internal forces used in the unassembled approach is generally of a high dimension. This makes the POD-DEIM reduction in itself computationally intensive, which is undesired. A second problem is that instability occurred in some of the POD-DEIM reduced responses. The reason for this instability is still unknown, but it is suspected to be related to the fact that the reductions done on the displacement field and the nonlinear part of the internal force are independent and thus have no common optimality criterion. The third important issue involves the application of POD-DEIM on finite elements that describe a combination of different types of internal forces and or moments. Snapshots of these so-called heterogeneous internal forces will have to be weighted before they can be used as an input to the POD.

order reduction
discrete
empirical
interpolation method

http://resolver.tudelft.nl/uuid:6f1531d7-a956-4c70-b8af-149111a9243d

2012-12-07

Student theses

master thesis

(c) 2012 Dedden, R.J.