[Paper Review] A discretize-then-map approach for the treatment of parameterized geometries in model order reduction

Tommaso Taddei, Lei Zhang|arXiv (Cornell University)|Oct 26, 2020

Model Reduction and Neural NetworksPhysics and Astronomy65 references15 citations

TL;DR

This paper introduces a discretize-then-map (DtM) approach for model order reduction (MOR) of parameterized PDEs with geometric variations, where the high-fidelity (HF) finite element (FE) mesh is first discretized on a reference domain, and then deformed via a parameterized mapping during the online stage. The method uses empirical quadrature to select a reduced set of elements, enabling efficient residual assembly using standard FE routines without modifying the HF code, and achieves high online efficiency and implementation simplicity compared to the standard map-then-discretize approach.

ABSTRACT

We present a general approach for the treatment of parameterized geometries in projection-based model order reduction. During the offline stage, given (i) a family of parameterized domains {Ωμ : μ ∈ P} ⊂ RD where μ ∈ P ⊂ RP denotes a vector of parameters, (ii) a parameterized mapping Φμ between a reference domain Ω and the parameter-dependent domain Ωμ, and (iii) a finite element triangulation of Ω, we resort to an empirical quadrature procedure to select a subset of the elements of the grid. During the online stage, we first use the mapping to “move” the nodes of the selected elements and then we use standard element-wise residual evaluation routines to evaluate the residual and possibly its Jacobian. We discuss how to devise an online-efficient reduced-order model and we discuss the differences with the more standard “map-then-discretize” approach (e.g., Rozza, Huynh, Patera, ACME, 2007); in particular, we show how the discretize-then-map framework greatly simplifies the implementation of the reduced-order model. We apply our approach to a two-dimensional potential flow problem past a parameterized airfoil, and to the two-dimensional RANS simulations of the flow past the Ahmed body.

Motivation & Objective

• Address the challenge of efficiently performing model order reduction (MOR) for parameterized PDEs on domains with geometric variations.
• Overcome limitations of the standard map-then-discretize (MtD) approach, which requires modifying HF solvers and may suffer from non-conforming meshes and complex variational formulations.
• Enable online-efficient reduced-order models that do not require changes to existing high-fidelity finite element codes.
• Simplify the implementation of ROMs—especially for nonlinear PDEs—by avoiding re-implementation of weak forms and coefficient terms.
• Ensure robustness and accuracy through empirical quadrature for residual evaluation and dual residual norm estimation for error control.

Proposed method

• Discretize the PDE on a reference domain Ω using a high-order isoparametric FE method before applying the parameterized mapping Φµ.
• Use an empirical quadrature (EQ) procedure during the offline stage to select a low-dimensional subset of elements (Ieq) for online residual evaluation.
• During the online stage, deform the selected elements' nodes via the mapping Φµ and reuse standard element-wise residual evaluation routines.
• Construct the reduced-order model using the deformed mesh and empirical quadrature weights to approximate the residual and its Jacobian efficiently.
• Enforce inhomogeneous Dirichlet boundary conditions via lifting functions (control function method), compatible with both strong and weak enforcement.
• Estimate the dual residual norm for error control using the empirical test space and quadrature weights, enabling online error estimation without solving the full HF problem.

Experimental results

Research questions

• RQ1 How can model order reduction be efficiently applied to parameterized geometries without modifying the high-fidelity finite element code?
• RQ2 What are the implementation and computational advantages of discretizing first and then mapping, compared to the standard map-then-discretize approach?
• RQ3 Can empirical quadrature be effectively used to reduce the computational cost of residual assembly in reduced-order models for parameterized geometries?
• RQ4 How can accurate online error estimation be achieved in the discretize-then-map framework using dual norms and empirical test spaces?
• RQ5 To what extent does the DtM approach maintain accuracy and convergence for nonlinear PDEs such as the RANS equations?

Key findings

• The discretize-then-map (DtM) approach enables online-efficient reduced-order models without requiring modifications to existing high-fidelity finite element solvers.
• Empirical quadrature reduces the number of elements used in residual assembly to less than 10% of the total mesh for the Ahmed body problem, achieving high computational efficiency.
• Dual residual norm estimation enables accurate online error control, with estimated residuals closely tracking the true relative H1 error in both the potential flow and RANS test cases.
• The method achieves high accuracy with a reduced integration domain (RID) that is not required to conform to the coarse-grained partition of the mapping, unlike the MtD approach.
• For the two-dimensional RANS simulation of the Ahmed body, the method successfully handles nonlinear, stabilized finite element formulations with SUPG, LSIC, and PSPG stabilization, demonstrating robustness for complex flows.
• The approach simplifies implementation—especially for nonlinear problems—by allowing reuse of standard element-wise residual routines without re-deriving weak forms with variable coefficients.