General treatment of essential boundary conditions in reduced order models for non-linear problems

Abstract

Inhomogeneous essential boundary conditions must be carefully treated in the formulation of Reduced Order Models (ROMs) for non-linear problems. In order to investigate this issue, two methods are analysed: one in which the boundary conditions are imposed in an strong way, and a second one in which a weak imposition of boundary conditions is made. The ideas presented in this work apply to the big realm of a posteriori ROMs. Nevertheless, an a posteriori hyper-reduction method is specifically considered in order to deal with the cost associated to the non-linearity of the problems. Applications to nonlinear transient heat conduction problems with temperature dependent thermophysical properties and time dependent essential boundary conditions are studied. However, the strategies introduced in this work are of general application.

Error estimation in reduced basis method for systems with time-varying and nonlinear boundary conditions

“…The ansatz formulation (6) cannot capture time-varying boundary conditions [17]. This is attributed to the global and time-invariant nature of the reduced basis vectors.…”

“…In the reduced-order model, different approaches for dealing with parameterized Dirichlet boundary conditions have been introduced [16,17]. In addition, the formulation of boundary conditions as a differential algebraic equation (DAE) has been discussed in [18].…”

The Pseudo-Direct Numerical Simulation Method considered as a Reduced Order Model

“…• The a posteriori methods are built after calculating solutions of the problem. The most classical example is the Proper Orthogonal Decomposition (POD) method [4][5][6][7], which is based on a statistical procedure called Principal Component Analysis (PCA). • The a priori methods are constructed without the need to compute any solution to the problem.…”

Model Order Reduction Technique for Large Scale Flow Computations

“…The second problem arises when you look closely at the nonlinearities in (6). One may notice that to evaluate the non-linearity in the reduced order model f (t, η ), it is necessary to evaluate the function f at (t, y ) and y (t) = j=1 η j (t)ψ j ∈ R m .…”

“…The presented reduced order model (ROM) creation technique represents an a posteriori approach to MOR [6]. Hence, the solution of the full order model (FOM) has to be available for the ROM creation.…”