CSRI Summer Proceedings 2022 3 THE SCHWARZ ALTERNATING METHOD FOR THE SEAMLESS COUPLING OF NONLINEAR REDUCED ORDER MODELS AND

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CSRI Summer Proceedings 2022 3

THE SCHWARZ ALTERNATING METHOD FOR THE SEAMLESS

COUPLING OF NONLINEAR REDUCED ORDER MODELS AND

FULL ORDER MODELS

JOSHUA BARNETT∗, IRINA TEZAUR , AND ALEJANDRO MOTA†

Abstract. Projection-based model order reduction allows for the parsimonious representation of

full order models (FOMs), typically obtained through the discretization of a set of partial diﬀerential

equations (PDEs) using conventional techniques (e.g., ﬁnite element, ﬁnite volume, ﬁnite diﬀerence

methods) where the discretization may contain a very large number of degrees of freedom. As

a consequence of this more compact representation, the resulting projection-based reduced order

models (ROMs) can achieve considerable computational speedups, which are especially useful in

real-time or multi-query analyses. One known deﬁciency of projection-based ROMs is that they can

suﬀer from a lack of robustness, stability and accuracy, especially in the predictive regime, which

ultimately limits their useful application. Another research gap that has prevented the widespread

adoption of ROMs within the modeling and simulation community is the lack of theoretical and

algorithmic foundations necessary for the “plug-and-play” integration of these models into existing

multi-scale and multi-physics frameworks. This paper describes a new methodology that has the

potential to address both of the aforementioned deﬁciencies by coupling projection-based ROMs

with each other as well as with conventional FOMs by means of the Schwarz alternating method

[41]. Leveraging recent work that adapted the Schwarz alternating method to enable consistent and

concurrent multi-scale coupling of ﬁnite element FOMs in solid mechanics [35, 36], we present a new

extension of the Schwarz framework that enables FOM-ROM and ROM-ROM coupling, following

a domain decomposition of the physical geometry on which a PDE is posed. In order to maintain

eﬃciency and achieve computation speed-ups, we employ hyper-reduction via the Energy-Conserving

Sampling and Weighting (ECSW) approach [9]. We evaluate the proposed coupling approach in the

reproductive as well as in the predictive regime on a canonical test case that involves the dynamic

propagation of a traveling wave in a nonlinear hyper-elastic material.

1. Introduction. Projection-based model order reduction is a promising data-

driven strategy for reducing the computational complexity of numerical simulations by

restricting the search of the solution to a low-dimensional space spanned by a reduced

basis constructed from a limited number of high-ﬁdelity simulations and/or physical

experiments/observations. While recent years have seen extensive investments in the

development of projection-based reduced order models (ROMs) and other data-driven

models, these models are known to suﬀer from a lack of robustness, stability and

accuracy, especially in the predictive regime. Moreover, a uniﬁed and rigorous theory

for integrating these models in a “plug-and-play” fashion into existing multi-scale and

multi-physics coupling frameworks (e.g., the Department of Energy’s Energy Exascale

Earth System Model (E3SM) [12]) is lacking at the present time.

This paper presents and evaluates an approach aimed at addressing both of the

aforementioned shortcomings by advancing the Schwarz alternating method [41] as

a mechanism for coupling together a variety of models, including full order ﬁnite el-

ement models and projection-based ROMs constructed using the proper orthogonal

decomposition (POD)/Galerkin method. The Schwarz alternating method is based on

the simple idea that if the solution to a partial diﬀerential equation (PDE) is known in

two or more regularly-shaped domains comprising a more complex domain, these local

solutions can be used to iteratively build a solution on the more complex domain. Our

coupling approach thus consists of several ingredients, namely: (1) the decomposition

of the physical domain into overlapping or non-overlapping subdomains, which can

be discretized in space by disparate meshes and in time by diﬀerent time-integration

∗Department of Mechanical Engineering, Stanford University, jb0@stanford.edu

†Sandia National Laboratories, ikalash@sandia.gov

arXiv:2210.12551v3 [math.NA] 10 Nov 2022

4The Schwarz Alternating Method for Coupling of Nonlinear ROMs and FOMs

schemes with diﬀerent time-steps, (2) the deﬁnition of transmission boundary condi-

tions on subdomain boundaries, and (3) the iterative solution of a sequence of subdo-

main problems in which information propagates between the subdomains through the

aforementioned transmission conditions. Without loss of generality, we develop and

prototype the method in the context of a generic transient dynamic solid mechanics

problem, deﬁned by an arbitrary constitutive model embedded within the governing

PDEs. Following an overlapping or non-overlapping domain decomposition (DD) of

the underlying geometry, we use the POD/Galerkin method with Energy-Conserving

Sampling and Weighting (ECSW)-based hyper-reduction [9] to reduce the problem in

one or more subdomains. We then employ the Schwarz alternating method to couple

the resulting subdomain ROMs with each other or with ﬁnite element-based full order

models (FOMs) in neighboring subdomains. We demonstrate that a careful formula-

tion and implementation of the transmission conditions in the ROMs being coupled

is essential to the coupling method.

The methodology described in this paper is related to several existing coupling

approaches developed in recent years. First, while this work is a direct extension of the

recently-developed Schwarz-based methodology for concurrent multi-scale FOM-FOM

coupling in solid mechanics [35, 36], it includes a number of advancements, including

the extension of the coupling framework to: (1) FOM-ROM coupling, (2) ROM-ROM

coupling, and (3) non-overlapping subdomains. Among the earliest authors to de-

velop an iterative Schwarz-based DD approach for coupling FOMs with ROMs are

Buﬀoni et al. [3]. The approach in [3] is unlike ours in that attention is restricted to

Galerkin-free POD ROMs, developed for the Laplace equation and the compressible

Euler equations. Other authors to consider Galerkin-free FOM-ROM and ROM-ROM

couplings are Cinquegrana et al. [4] and Bergmann et al. [2]. The former approach

[4] focuses on overlapping DD in the context of a Schwarz-like iteration scheme, but,

unlike our approach, requires matching meshes at the subdomain interfaces. The lat-

ter approach [2], termed zonal Galerkin-free POD, deﬁnes a minimization problem

to minimize the diﬀerence between the POD reconstruction and its corresponding

FOM solution in the overlapping region between a ROM and a FOM domain, and is

developed/investigated in the context of an unsteady ﬂow and aerodynamic shape op-

timization. While the method developed [2] is not based on the Schwarz alternating

formulation, the recent related work [22] by Iollo et al. demonstrates that a sim-

ilar optimization-based coupling scheme is equivalent to an overlapping alternating

Schwarz iteration for the case of linear elliptic PDE. A true POD-Greedy/Galerkin

non-overlapping Schwarz method for the coupling of projection-based ROMs devel-

oped for the speciﬁc case of symmetric elliptic PDEs is presented by Maier et al. in

[34].

While the focus herein is restricted to projection-based ROMs, it is worth not-

ing that the Schwarz alternating method has recently been extended to the case of

coupling Physics-Informed Neural Networks (PINNs) to each other following a DD in

[28, 29]. The methods proposed in these works, termed D3M [28] and DeepDDM [29],

inherit the beneﬁts of DD-based ROM-ROM couplings, but are developed primarily

for the purpose of improving the eﬃciency of the neural network training process and

reducing the risk of overﬁtting, both of which are due to the global nature of the

neural network “basis functions”.

We end our literature overview by remarking that a number of non-Schwarz-

based ROM-ROM and/or FOM-ROM coupling methods have been developed in re-

cent years, including [1, 46, 15, 20, 27, 32, 33, 5, 6, 24, 23, 39, 8, 43, 40, 18]. The

J. Barnett, I. Tezaur and A. Mota 5

majority of these approaches are based on Lagrange multiplier or ﬂux matching cou-

pling formulations, and focus on either simple linear elliptic PDEs or ﬂuid problems.

We omit a detailed assessment of these references from this paper for the sake of

brevity.

The remainder of this paper is organized as follows. In Section 2, we provide

the variational formulation of the generic solid dynamics problem considered herein,

and describe its spatio-temporal discretization. Section 3 details our nonlinear model

reduction methodology for this problem, which relies on the POD/Galerkin approach

for model reduction and the ECSW method [9] for hyper-reduction. We describe the

Schwarz alternating method for FOM-FOM, FOM-ROM and ROM-ROM coupling in

Section 4. In Section 5, we evaluate the performance of the proposed Schwarz-based

coupling methodology on a problem involving dynamic wave propagation in a one-

dimensional (1D) hyper-elastic bar whose material properties are described by the

nonlinear Henky constitutive model, characterized by a logarithmic strain tensor [14].

We conclude this paper with a summary and a discussion of some future research

directions (Section 6).

2. Solid mechanics problem formulation. Consider the Euler-Lagrange equa-

tions for a generic dynamic solid mechanics problem in its strong form:

Div P+ρ0B=ρ0¨

ϕin Ω ×I. (2.1)

In (2.1), Ω ∈Rdfor d∈ {1,2,3}is an open bounded domain, I:= {t∈[0, T ]}is

a closed time interval with T > 0, and x=ϕ(X, t) : Ω ×I→Rdis a mapping,

with X∈Ω and t∈I. The symbol Pdenotes the ﬁrst Piola-Kirchhoﬀ stress and

ρ0B: Ω →Rdis the body force, with ρ0denoting the mass density in the reference

conﬁguration. The over-dot notation denotes diﬀerentiation in time, so that ˙

ϕ:= ∂ϕ

∂t

and ¨

ϕ:= ∂2ϕ

∂t2. Embedded within Pis a constitutive model, which can range from a

simple linear elastic model to a complex micro-structure model, e.g., that of crystal

plasticity. Herein, we focus on nonlinear hyper-elastic constitutive models such as the

Henky model [14]. The details of this model are provided in Section 5.

Suppose that we have the following initial and boundary conditions for the PDEs

(2.1):

ϕ(X, t0) = X0,˙

ϕ(X, t0) = v0in Ω,

ϕ(X, t) = χon ∂Ωϕ×I, P N =Ton ∂ΩT×I. (2.2)

In (2.2), it is assumed the outer boundary ∂Ω is decomposed into a Dirichlet and

traction portion, ∂Ωϕand ∂ΩT, respectively, with ∂Ω = ∂Ωϕ∪∂ΩTand ∂Ωϕ∩

∂ΩT=∅. The prescribed boundary positions or Dirichlet boundary conditions are

χ:∂Ωϕ×I→R3. The symbol Ndenotes the unit normal on ∂ΩT. In this work,

we will assume without loss of generality that χis not changing in time.

It is straightforward to show that the weak variational form of (2.1) with initial

and boundary conditions (2.2) is

ZIZΩ

(Div P+ρ0B−ρ0¨

ϕ)·ξdV+Z∂TΩ

T·ξdSdt= 0,(2.3)

where ξis a test function in V:= ξ∈W1

2(Ω ×I) : ξ=0on ∂ϕΩ×I∪Ω×t0∪

Ω×t1}.

6The Schwarz Alternating Method for Coupling of Nonlinear ROMs and FOMs

Discretizing the variational form (2.3) in space using the classical Galerkin ﬁnite

element method (FEM) [19] yields the following semi-discrete matrix problem:

M¨

u+fint(u,˙

u) = fext.(2.4)

In (2.4), Mdenotes the mass matrix, u:= ϕ(X, t)−Xis the displacement, ¨

uis

the acceleration (also denoted by a), fext is a vector of applied external forces, and

fint(u,˙

u) is the vector of internal forces due to mechanical and other eﬀects inside

the material, where ˙

u(also denoted by v) is the velocity. In the present work, the

semi-discrete equation (2.4) is advanced forward in time using the Newmark-βtime-

integration scheme [37]. We will assume the FOM (2.4) has size N∈N, that is, u∈

RN. For convenience, in subsequent discussion, we transform the Dirichlet boundary

condition (2.2) for the position into a Dirichlet boundary for the displacement, so that

the boundary condition imposed on ∂Ωϕ×Iis u=uD, with uDbeing independent

of time.

3. Model order reduction. Projection-based model order reduction is a promis-

ing, physics-based technique for reducing the computational cost associated with high-

ﬁdelity models such as (2.4). The basic workﬂow for building a projection-based ROM

for a generic nonlinear semi-discrete problem of the form (2.4) consists of three steps:

(1) calculation of a reduced basis, (2) projection of the governing equations onto the

reduced basis, and (3) hyper-reduction of the nonlinear terms in the projected equa-

tions. Herein, we employ the POD [42, 16] for the reduced basis generation step (step

1), the Galerkin projection method for the projection step (step 2), and the ECSW

method [9] for the hyper-reduction step (step 3). Each of these steps is described

succinctly below.

3.1. Proper orthogonal decomposition (POD). The POD is a mathemat-

ical procedure that, given an ensemble of data and an inner product, constructs a

basis for the ensemble that is optimal in the sense that it describes more energy (on

average) of the ensemble in the chosen inner product than any other linear basis

of the same dimension M. The ensemble ws∈RN:s= 1, ..., S is typically a set of

Sinstantaneous snapshots of a numerical solution ﬁeld, collected for Svalues of a

parameter of interest, and/or at Sdiﬀerent times. For solid mechanics problems, a

natural choice for the snapshots is ws=us, where the ensemble {us}denotes a set

of snapshots for the displacement ﬁeld. It is noted that one can use in place of on

addition to {us}snapshots of the velocity ({vs}) and/or acceleration ({as}) ﬁelds.

Following the so-called “method of snapshots” [42, 16], a POD basis ΦM∈RN×M

of dimension Mis obtained by performing a singular value decomposition (SVD) of

a snapshot matrix W:= w1, ..., wS∈RN×Ssuch that W=ΦΣVTand deﬁning

ΦMas the matrix containing the ﬁrst Mcolumns of Φ. Letting

EPOD(M) := PM

i=1 σ2

(3.1)

denote the energy associated with a POD basis of size M, where σidenotes the ith

singular value of W, the basis size Mis typically selected based on an energy criterion,

where 100EPOD(M) is the percent energy captured by a given POD basis.

As discussed in Section 3.2, it is often desirable to construct the POD basis ΦM

such that this basis satisﬁes homogeneous Dirichlet boundary conditions at some pre-

deﬁned indices id∈Ndwhere d < N. It is straightforward to accomplish this by

J. Barnett, I. Tezaur and A. Mota 7

simply zeroing out the snapshots at the Dirichlet degrees of freedom (dofs) prior to

calculating the SVD, that is, setting wi(id) = 0for i= 1, ..., S.

3.2. Galerkin projection. As discussed in [9, 45], Galerkin projection is in

general the method of choice for solid mechanics and structural dynamics problems,

as it preserves the Hamiltonian structure of the underlying system of PDEs [26]. The

method starts by approximating the FOM displacement solution to (2.4) as

u≈˜

u=¯

u+ΦMˆ

u,(3.2)

where ˆ

u∈RMis the vector of unknown modal amplitudes, to be solved for in the

ROM and ¯

u∈RNis a (possibly time-dependent) reference state. Similar approx-

imations can be made for the velocity and acceleration ﬁelds, v:= ˙

uand a:= ¨

respectively, namely:

v≈˜

v=¯

v+ΦMˆ

a≈˜

a=¯

a+ΦMˆ

a.(3.3)

Here, ¯

vand ¯

aare deﬁned analogously to ¯

u, and similarly for ˆ

vand ˆ

In the present formulation, the reference states ¯

u,¯

vand ¯

aare used to prescribe

strongly Dirichlet boundary conditions within the ROM for the displacement, ve-

locity and acceleration ﬁelds, respectively. This is done following the approach of

Gunzburger et al. [13]. Suppose the Dirichlet boundary conditions of interest are

u(id) = uD,v(id) = vDand a(id) = aD. Let iudenote the unconstrained indices

at which the solution to (2.4) is sought. It is straightforward to see from (3.2)–(3.3)

that, if the POD modes ΦM= [φ1, ..., φM] are calculated such that φi(id) = 0for

i= 1, ..., M and

u(id) = uD,¯

u(iu) = 0,

v(id) = vD,¯

v(iu) = 0,

a(id) = aD,¯

a(iu) = 0,

(3.4)

the ROM solution will satisfy the prescribed Dirichlet boundary conditions in the

strong sense.

The ROM for (2.4) is obtained by substituting the decompositions (3.2)–(3.3)

into the FOM equations (2.4), and projecting these equations onto the reduced basis

ΦM. It is straightforward to verify that doing this and moving all Dirichlet dofs to

the right-hand side of the resulting system of equations yields a semi-discrete problem

of the form

Muu ˆ

a+ˆ

fint

u(˜

u,˜

v) = ˆ

fext

u,(3.5)

where

Muu := ΦT

M,uMuuΦM,u,fint

u(˜

u,˜

v) := ΦT

M,ufint

u(˜

u,˜

v),

fext

u:= ΦT

M,u(fext

u−Mud ¯

ad).(3.6)

In (3.6), ΦM,u := ΦM(iu,:), Muu := M(iu,iu), fint

u:= fint(iu), fext

u:= fext(iu),

Mud := M(iu,id) and ¯

ad:= ¯

a(id).

Remark 1. It is noted that, while all three variables ˜

u,˜

vand ˜

aappear in the

ROM system (3.6), we are not considering the displacement, velocity and acceleration

variables as independent ﬁelds with separate generalized coordinates in our ROM

construction approach. In particular, ˆ

v:= dˆ

dt and ˆ

a:= d2ˆ

dt2in (3.3). Taking this

approach ensures consistency between the displacement, velocity and acceleration

solutions computed within the ROM.

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CSRISummerProceedings20223THESCHWARZALTERNATINGMETHODFORTHESEAMLESSCOUPLINGOFNONLINEARREDUCEDORDERMODELSANDFULLORDERMODELSJOSHUABARNETT,IRINATEZAUR,ANDALEJANDROMOTAyAbstract.Projection-basedmodelorderreductionallowsfortheparsimoniousrepresentationoffullordermodels(FOMs),typicallyobtainedthroughthed...

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