Robust and globally efficient reduction of parametric, highly nonlinear computational models and real time online performance

Abstract

This paper considers the problem of parametric, nonlinear, projection-based model order reduction (PMOR) and makes three contributions to advance its state of the art. First, it presents a computational strategy for reducing the total offline cost of nonlinear PMOR by enabling hyperreduction to be performed adaptively within the typical greedy sampling procedure, rather than uniformly at each of its iteration. Then, it presents a more accurate and computationally efficient approach for training hyperreduction based on residual Jacobians rather than residuals. Finally, it demonstrates for a large-scale, industrial-grade, parametric, nonlinear application, that the two aforementioned contributions make both offline and online stages of nonlinear PMOR practical in today’s engineering environments. The paper also shows that despite the so-called Kolmogorov $n$-width barrier for the model reduction of convection-dominated transport problems, the current state of the art of nonlinear, Petrov–Galerkin PMOR equipped with the advances proposed in this paper is capable of accelerating by orders of magnitude the accurate solution of shape-parametric, high Reynolds number, viscous flow problems represented by the Reynolds-averaged Navier–Stokes equations augmented with turbulence modeling. All three contributions are presented and discussed in the contexts of steady-state CFD problems; however, they are equally applicable to unsteady CFD problems as well as problems in solid mechanics and structural dynamics.

Introduction

While computational models in structural dynamics, fluid dynamics, and other engineering and scientific fields can circumvent the need for expensive experiments such as frontal barrier impact and wind tunnel tests, the computational cost of fully resolved numerical models is often intractable. Modeling simplifications and assumptions such as, in the case of computational fluid dynamics (CFD), turbulence and near-wall modeling or inviscid and incompressible flows, can decrease the computational cost without compromising accuracy — when performed appropriately. However, the computational cost of high-fidelity, high-dimensional models (HDMs) often remains prohibitive for many parametric applications such as the modeling and quantification of model-form uncertainty [1], routine analysis, and design optimization. Projection-based model order reduction (PMOR) is a promising computational technology for enabling such applications, starting from a parametric HDM of interest whose dimension is denoted throughout this paper by $N$. It creates surrogate models that take advantage of the lower dimensionality of the manifold of the HDM-based solutions. Specifically, it produces projection-based reduced-order models (PROMs) of much lower dimension $n\ll N$. Unlike regression-based approaches for constructing data fitting surrogate models, a PMOR method incorporates the physics underlying a HDM by projecting it onto a subspace of much smaller dimension, using typically an affine approximation based a properly trained reduced-order basis $\mathbf{V}\in {\mathbb{R}}^{N\times n}$: this enables it to deliver accurate predictions for any quantity of interest (QoI) not explicitly accounted for during training. In order to capitalize on the lower dimensionality of a subspace and translate it into a lower computational cost, including in the parametric setting, hyperreduction is often invoked for approximating projected (or reduced) quantities. In this case, a nonlinear PROM is transformed into a computationally efficient and yet accurate hyperreduced PROM (HPROM). This paper presents a methodology for constructing a “globally efficient” HPROM for parametric, nonlinear, static or steady-state applications in computational mechanics, that is practical in engineering environments. For this purpose, it focuses on the reduction of the offline costs associated with an iterative, adaptive procedure for constructing a nonlinear HPROM; the enhancement of the accuracy and computational efficiency of the hyperreduction training; the proper scaling of a nonlinear HPROM; and on the efficient initialization of a Newton-like solution procedure during the online execution of a nonlinear, implicit, discrete HPROM.

Usually, the computational complexity of the construction of a parametric PROM of dimension $n$ scales with both $n$ and the larger dimension $N\gg n$ of the underlying HDM. For the class of parametric, linear HDMs admitting an efficient parameter-affine representation and that of parametric or nonparametric nonlinear HDMs with a low-order polynomial dependence on the solution, this issue can be addressed by precomputing those reduced-order quantities whose computational complexities scale with $N$ [2], [3], [4], [5]. Lifting transformations [6] can sometimes be used to treat the case of nonpolynomial nonlinearities (for example, see [7]), but hyperreduction [5] is typically required in all other cases to achieve a computational complexity that, in the worst case, scales as $\mathcal{O}\left(n\right)$ during the online evaluation of the reduced-order quantities.

One class of hyperreduction methods described in [5] as methods of the approximate-then-project type, originated in image-reconstruction algorithms such as the gappy proper orthogonal decomposition (POD) method [8]. They first approximate a nonlinear high-dimensional quantity, for example, by interpolating a subset of carefully selected degrees of freedom, then project it on the subspace represented by the right ROB $\mathbf{V}$. Examples include the empirical interpolation method (EIM) [9], the collocation method [10], and the discrete empirical interpolation method (DEIM) [11] – which is arguably the most popular hyperreduction method of this type. The other class of hyperreduction methods, described in [5] as methods of the project-then-approximate type, have been more recently developed: they seek to approximate directly the reduced-order quantities. Often, they can be interpreted as generalized quadrature rules. They include the cubature-based approach developed for computational graphics [12], the energy-conserving sampling and weighting (ECSW) method [13], the empirical cubature method (ECM) [14], and the linear program-based empirical quadrature method [15]. ECSW, is adopted in this paper for the following three reasons. Firstly, it features for Galerkin PROMs associated with second-order dynamical systems such as those that arise in structural dynamics, unique and attractive numerical stability properties that have been demonstrated at both the theoretical and numerical levels, for large-scale applications [16]. Secondly, ECSW has been recently extended in [17] to Petrov–Galerkin PROMs — which have been shown to demonstrate in practice and for many applications better numerical stability properties than their Galerkin counterparts [18], [19], [20]. In particular, ECSW was extended in [17] to nonlinear PROMs generated by the least squares Petrov–Galerkin (LSPG) PMOR method, which is notorious for its robustness and superior performance for convection dominated turbulent flow problems [20], [21]; and achieved for a large-scale $\left(N=\mathcal{O}\left(10^8\right)\right)$ turbulent flow application of industrial relevance three and five orders of magnitude speedup factors for the wall-clock and CPU solution times.

To achieve robustness in a parameter domain ${\Omega }_{\mu }$, the ROB $\mathbf{V}$ underlying a nonlinear PROM is usually trained in this domain using an iterative parameter sampling procedure known as a greedy procedure. At each iteration, such a procedure adaptively selects the next parameter point to sample in ${\Omega }_{\mu }$, based on feedback information generated by an error indicator; exercises the parametric HDM at the sampled parameter point to compute at this point one or several solution snapshots; and then rebuilds the ROB $\mathbf{V}$. The achieve computational tractability, at each iteration, the error indicator needs to approximate the HDM-based solution at number of different test points in ${\Omega }_{\mu }$. Whether these are preselected or randomly regenerated at each iteration, computational tractability calls for computing the approximate solutions used by the error indicator using an HPROM. For this reason, a greedy sampling procedure entails in general repetitive rebuilds of an HPROM, which increases the offline computational cost of PMOR. A first contribution of this paper is the reduction of that cost by enabling rebuilding an HPROM only as needed.

As proposed in [13], [17], the hyperreduction method ECSW is itself trained with the residuals of the solution snapshots computed during a greedy sampling procedure, or a subset of them. However, in many circumstances that are identified in this paper, these residuals are close to zero, in which case ECSW is poorly trained. Hence, a second contribution of this paper is the more accurate and computationally efficient training of ECSW using instead the Jacobians of the residuals (which are readily available in implicit computations).

A third contribution of this paper is the demonstration for a large-scale, industrial-grade, parametric, nonlinear application, that the two aforementioned contributions equipped with of a proper scaling procedure for the construction of a nonlinear, implicit, discrete HPROM and an efficient initialization of a Newton-like solution procedure during its online execution lead to a globally efficient approach for performing nonlinear PMOR — that is, an approach that makes both offline and online stages of PMOR practical in today’s engineering environments.

Due to space limitation and for the reasons highlighted above, the contributions of this paper are presented in the contexts of steady-state CFD problems, the nonlinear PMOR method LSPG, and the hyperreduction method ECSW. However, they are equally applicable, in principle, to unsteady CFD problems and many other areas of computational mechanics including computational structural and solid mechanics problems, and using many other PMOR and hyperreduction methods.

The remainder of this paper is organized as follows. Section 2 formalizes the contexts specified above in order to keep this paper as self-contained as possible. Section 3 revisits the hyperreduction method ECSW, presents a better approach for training it based on residual Jacobians, and describes a computational strategy for reducing its overall cost when implemented within a greedy sampling procedure. Section 4 demonstrates all contributions of this paper for two different CFD applications: a relatively simple one that has the merit of being easily reproducible by the interested reader; and an industrial-grade CFD application that highlights the merits of the proposed advancements of the state of the art of parametric, nonlinear PMOR. Finally, Section 5 concludes this paper.