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Geometric learning for computational mechanics Part III. Physics-constrained response surface of geometrically nonlinear shells

June 2023
Computer Methods in Applied Mechanics and Engineering 415(1)

DOI:10.1016/j.cma.2023.116219

Authors:

Mian Xiao

Columbia University

Ran Ma

Southeast University

Waiching Sun

Columbia University

This paper presents a graph-manifold iterative algorithm to predict the configurations of geometrically exact shells subjected to external loading. The finite element solutions are first stored in a weighted graph where each graph node stores the nodal displacement and nodal director. This collection of solutions is embedded onto a low-dimensional latent space through a graph isomorphism encoder. This graph embedding step reduces the dimensionality of the nonlinear data and makes it easier for the response surface to be constructed. The decoder, in return, converts an element in the latent space back to a weighted graph that represents a finite element solution. As such, the deformed configuration of the shell can be obtained by decoding the predictions in the latent space without running extra finite element simulations. For engineering applications where the shell is often subjected to concentrated loads or a local portion of the shell structure is of particular interest, we use the solutions stored in a graph to reconstruct a smooth manifold where the balance laws are enforced to control the curvature of the shell. The resultant computer algorithm enjoys both the speed of the nonlinear dimensional reduced solver and the fidelity of the solutions at locations where it matters.

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Computer Methods in Applied Mechanics and Engineering manuscript No.

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Geometric learning for computational mechanics Part III.1

Physics-locally-constrained digital twin of geometric nonlinear shells2

Mian Xiao ·Ran Ma ·WaiChing Sun3

Received: April 19, 2023/ Accepted: date5

Abstract This paper presents a graph-manifold iterative algorithm to predict the conﬁgurations of shells6

subjected to external loading. We ﬁrst stored ﬁnite element solutions of stress resultant geometric exact7

shell of Simo, Fox & Rafai, 1990 in a weighted graph of which the nodes store the nodal, displacement8

and reconstructed Gaussian curvature. This collection of solutions are then embedded into latent space9

through a graph isomorphism autoencoder. The low-dimensionality and the Euclidean properties of the10

latent space enable us to construct response surfaces efﬁciently, while the decoder enables us to map the11

encoded latent solution back directly to the decoded geometrically exact deformed conﬁguration of the12

shell discretized by ﬁnite elements. For engineering applications where the shell is often subjected to con-13

centrated loads or a local portion of the shell structure is of particular interest, we use the solutions stored14

in a graph to reconstruct a smooth manifold where the balance laws are enforced to control the curvature15

of the shell. The resultant computer algorithm enjoys both the speed of the nonlinear dimensional reduced16

solver, and the ﬁdelity of the solutions at locations where it matters.17

Keywords machine learning, graph neural network, shell, reduced order modeling, curvature control18

1 Introduction19

Shells are structural elements commonly found in numerous applications, such as the roof of a building20

[8,30,39,54], the wings of airplanes and space structures [41], wind turbine blades [5], as well as deploy-21

able structures like parachutes [2] and papers [6]. In fact, shell structures all share one geometric feature –22

the kinematics of the mid-surface of the structure is sufﬁcient to represent the kinematics of the deformed23

conﬁgurations. As such, while points on the mid-surface can be considered as elements of a Riemannian24

manifold that lacks global coordinate, ﬁnite element discretization may provide us with a systematic way25

to represent the geometry of the mid-surface as a collection of connecting two-dimensional shell elements26

with corners deﬁned in a three-dimensional Euclidean space.27

To enforce the dimensional constraint in a ﬁnite element shell model, the degenerated solid approach [13,28

24,37,50] is widely adopted in the early 80s. Meanwhile, capturing the geometrical nonlinearity becomes29

necessary for shells that undergo signiﬁcant deformation to maintain the accuracy of predictions. In those30

cases, geometrically consistent shell models, such as [15,42], are often used such that the balance laws are31

parametrized in a way that avoids the explicit appearance of the Riemannian connection of the mid-surface32

but capture the geometrical effects (cf. l Ibrahimbegovi´

c [25], Roh and Cho [47], Simo et al. [48], Simo and33

Fox [49]).34

A key technical barrier in computational modeling of shell is to effectively represent the shell con-35

ﬁguration in the geometrically nonlinear regime with a singularity-free parametrization with the surface36

Corresponding author: WaiChing Sun

Associate Professor, Department of Civil Engineering and Engineering Mechanics, Columbia University, 614 SW Mudd, Mail

Code: 4709, New York, NY 10027 Tel.: 212-854-3143, Fax: 212-854-6267, E-mail: wsun@columbia.edu

2 Mian Xiao et al.

displacement and director expressed relative to an intrinsic frame. Simo and Fox [49] resolved this issue by37

idealizing a shell as a Riemannian manifold, a subclass of differentiable manifold with Euclidean tangential38

spaces [28]. Apart from shell structures, Riemannian manifold has been used to represent a large variety39

of objects and defects of materials commonly encountered in engineering applications, including but not40

limited to ﬁlms [34,56], crack surfaces [36,43], and lower-dimensional substructures in metamaterials or41

composites [17,27,32,51].42

Notice that while geometrically exact shell ﬁnite elements may provide an improvement in computa-43

tional efﬁciency over conventional 3D continuum ﬁnite element in the ﬁnite deformation regime, many44

applications of the shell, such as those used in generating vehicle crashing tests or response surface, of-45

ten requires a proper dimensional reduction strategy to generate a large number of numerical solutions46

within limited time. However, as the nonlinearity could be introduced by either or both the material and47

geometrical nonlinearities, classical modal analysis based on eigenvalue decomposition is not strictly ap-48

plicable. Alternatively, orthogonal bases can be obtained via principal component analysis, singular value49

decomposition [16,33]. These orthogonal bases are then used to construct Galerkin projection to reduce50

the dimensionality of the system of equations [26]. In all of these cases, the Galerkin projection can be con-51

sidered a linear embedding of the solution. As such, the quality of the predictions often depends on the52

choices of the orthogonal basis used to perform the projection [31], which may require a large data set for53

sampling and training [26]. Such a dataset for generating the orthogonal bases (through, for instance, the54

method of snapshot [10,65,66]) could sufﬁciently populate the parametric space. But in this sense, this55

dataset may contain sufﬁcient information to generate a solution manifold, so that without any further full-56

scale or reduced-order simulations, we may obtain any element within a class of solution with respect to57

a parametric domain (say different tractions applied at the same area of the shell). This solution manifold58

can be regarded as a hypersurface in a (N+P)-dimensional Euclidean space where Nis the dimensions59

of the ﬁnite element space, and Pis the total number of parameters necessary to parametrize the external60

loading. Such a solution manifold can then serve as the digital twins where the solution manifold may61

directly predict the responses of the shell structures from a speciﬁc class of external loadings.62

This paper is the Part III of the geometric learning for computational mechanics series in which we con-63

tinue to leverage geometric learning techniques showcased in Vlassis et al. [59] and Vlassis and Sun [58] for64

solid mechanics problems. The purpose of this paper is to build such a digital twin model for forecasting65

behaviors of geometrical nonlinear shell. To simplify the problem, we restrict our attention to the prob-66

lems where bifurcations or instability, such as buckling, do not occur [38]. This assumption eliminates the67

branching of the solution and ensures that each deformed conﬁguration corresponds to a particular exter-68

nal loading represented as a point in the parametric space. We then use a predictor-corrector algorithm to69

generate the solution manifold/hypersurface in a (N+P)-dimensional Euclidean space. In the predictor70

step, we ﬁrst reduce the complexity of the learning problems by approximating the ﬁnite element solu-71

tion of Simo-Fox-Rafai shell as a weighted graph. This graph representation in return provides us another72

opportunity to perform nonlinear dimensional reduction through graph embedding. As the graph embed-73

ding maps each solution graph onto a point at the latent Euclidean space, we then use neural networks to74

create a hypersurface in a (N+P)-dimensional Euclidean space where all the solution data is supposed75

to be lied on. In the corrector step, we introduce corrections to ensure the admissibility of the solution76

predicted by the neural network. This correction is done by applying (1) a cost-efﬁcient curvature ﬁlter to77

control the smoothness and curvature. and (2) a more computationally intensive physics-informed correc-78

tion that re-parametrizes a small domain of interest such that enforcing the balance principles would not79

lead to a difﬁcult high-dimensional non-convex optimization problem that often attributed to the failures80

of some physics-informed neural network in the literature (cf. Krishnapriyan et al. [29]).81

The rest of the paper is organized as follows: Section 2presents a detailed graph representation of the82

shell manifold and, followingly the graph-manifold learning formulation. Section 3introduces the curva-83

ture constraints and physical governing equations to enforce with implementation perspectives. Section 484

shows two numerical examples with problem conﬁgurations to demonstrate how the dataset is collected85

from ﬁnite element simulation results. Section 5lists prediction results for the two examples of interest that86

justify the correctness and applicability of the proposed geometric learning model. Section 6summarizes87

the conclusions for this paper. Section 7acknowledges the sponsors of the authors.88

Geometric learning for shells 3

2 Geometric learning with graph neural networks89

In this section, we explain how we use the graph embedding algorithm to extract a low-dimensional90

embedding for the node-weighted graphs storing the deformed conﬁgurations of shell structures obtained91

from varying a set of loading parameters. In Section 2.1, we explain the interpretation of the deformed92

shell ﬁelds as an undirected node-weighted graph. In Section 2.2, we demonstrate how the graph autoen-93

coder embeds the deformed shell conﬁgurations onto a lower-dimensional latent space for reduced-order94

predictions through the graph isomorphism layers.95

2.1 Uniﬁed graph representation for shells of different meshes96

To ensure the robustness of the digital twin without exhausting our resources for high-ﬁdelity sim-97

ulations, we introduce a data augmentation technique in which we may run a limited amount of high-98

ﬁdelity simulations (or experimental tests) for the important parametric domains, while utilizing more99

cost-efﬁcient low-ﬁdelity simulations (in our cases, the same ﬁnite element simulations with a coarser100

mesh) to populate the less critical regions of the parametric domain.101

We consider ﬁnite element nodes as graph vertices and form edges by considering how these nodes102

are related [58]. As such, nodal solutions can be viewed as the weights of the node. Assuming that we103

only use one type of testing and interpolating functions for all ﬁnite elements, then there is no need to104

introduce weights on the edges to distinguish the interpolations. As such, feasible choices of representing105

ﬁnite element solutions in a mesh include node graph, dual graph, communication graph [45] as well as106

tree [40]. In this work, we speciﬁcally adopt the node graph representation, which considers nodes in the107

same ﬁnite element as a one-hop neighborhood.108

In the following part of this section, we will introduce the mathematical expression of ﬁnite element109

node graph as a foundation for our machine learning model.110

A ﬁnite element node graph is an undirected graph G= (V,E), where V={vi|i=1, ..., N}is the111

node set of the graph as vicorresponding to individual ﬁnite element nodes, and E={(v1j,v2j)|j=112

1, ..., M;v1j,v2j∈ V} is the edge set where the existence of each individual edge indicates that node v1j

113

and v2jbelong to the same element, as shown in Fig. 1.Nand Mindicate the size of the node set and edge114

set. In order to deal with geometrical features to be incorporated in the machine learning model, we enrich115

the graph representation as a node-weighted graph: G0= (V,E,X)where X={xi∈RD|i=1, ..., N}is116

the nodal feature set as xiindicates the geometrical feature vector at node viwith a dimension of D. We117

may enforce D≥3 as the manifold reconstruction task requires at least the three spatial coordinates of the118

ﬁnite element point cloud.119

Fig. 1: Node graph interpretation of a ﬁnite element mesh.

With ﬁnite element node graph deﬁned, the fundamental geometric learning task for the reduced order120

modeling on the shell can be stated as follows: given a series of snapshots of the deformed shell manifold121

of the same initial conﬁguration and path-independent material properties G={G0

1, ..., G0

T}sampled by a122

subset of the parametrized load p∈RP, we would like to predict the shell manifold geometry G0deformed123

by any element of p∈RPvia the reduced-order latent space of G.124

4 Mian Xiao et al.

In the geometric learning literature, the embedding that performs the feature engineering or nonlinear125

dimensional reduction is often referred to as the upstream task, whereas the predictions that leverage the126

feature to make predictions is called the downstream application [7,9]. In our case, we follow the same127

workﬂow. We ﬁrst embed these shell manifold snapshots represented by node-weighted graphs onto a low128

dimensional latent space RDenc (upstream task), then we learn how pgoverns the shell conﬁguration not129

via the graph or the original ﬁnite element space, but in the low-dimensional latent space RDenc , where130

Denc Nis the dimension of the latent space.131

To establish a trade-off between ﬁdelity and cost efﬁciency, the collection of snapshots in Gthat consti-132

tute the database for training the digital twins may come from ﬁnite element meshes of different resolu-133

tions. In our numerical experiments, we collect data from two ﬁnite element meshes, i.e.,134

–High-ﬁdelity set: hG={hG0

1, ..., hG0

Th}.135

–Low-ﬁdelity set lG={lG0

1, ..., lG0

Tl}136

where ThTland hG∩lG=∅and hG∪lG=G.137

While the message-passing GIN can, in theory, generate embedding for graphs of different number of138

nodes, our interest is on predicting the ﬁnite element solution of the ﬁne-mesh solution without actually139

running the full-scale simulations. Hence, we introduce a new mechanism into the graph autoencoder140

architecture such that the reconstructed graph is corresponding to the ﬁne mesh used to generate hG,141

whereas a nonlinear mapping that maps the node features of the coarse-mesh graphs onto those of the142

ﬁne-mesh graphs is learned within the autoencoder before the latent space is learned. This design can143

break down into three subtasks, i.e.,144

1. Generate a richer high ﬁdelity set h˜

G={h˜

1, ..., h˜

Tl}, such that h˜

j∼lG0

j.145

146

2. Perform graph embedding based on the combined high ﬁdelity set h˜

G.147

148

3. Learn the governing dynamics that maps pto the lower-ordered latent space.149

where ∼indicates that h˜

jis the high ﬁdelity result on the ﬁner mesh corresponding to the low ﬁdelity150

result lG0

jof the same loading condition. For simplicity, we assume that all snapshots in hGcome from151

results on the same ﬁne mesh, while all snapshots in lGcome from results on the same mesh that is much152

coarser than their ﬁne counterpart.153

Remark 1 We may efﬁciently store the information of a node-weighted graph in a matrix tuple (A,X),154

which is convenient to use as the neural network input in the following sections. Ais called the adjacency155

matrix of size N×N, where Aij =Aji =1 while edge (v1,vj)exists otherwise Aij =Aji =0. Xis156

called the nodal feauture matrix of size N×Dsuch that the i-th row of Xcorrespond to xi. We notice157

that as the shell conﬁguration deformed during ﬁnite element simulation without remeshing, the node158

and edge connection remain the same, indicating the adjacency matrix of the resultant node graphs does159

not change. As such, all snapshots in lGhas the same adjacency matrix Aland all snapshots in hGhas the160

same adjacency matrix Ah. The converted snapshots in h˜

Gshould also have the same adjacency matrix as161

Ah.162

2.2 The geometric learning model with graph neural networks163

This section demonstrates how we resolve the aforementioned geometric learning subtasks. We ﬁrst164

introduce the neural network architecture used in Section 2.2.1. Then, we present the formulation of these165

learning tasks: Section 2.2.2 shows how we learn the mapping for building h˜

Gand then ﬁnd the reduced166

ordered latent space; Section 2.2.3 shows how predicts the shell geometry with pbased on the latent space.167

A sketch of the modeling procedures in this section is summarized in Fig 2.168

Geometric learning for shells 5

Fig. 2: Procedure sketch of the proposed geometric model. Notice that the portion number (90% low-ﬁdelity

data and 10% high-ﬁdelity data) in step 1 is not strict but related to the capacity to acquire high-ﬁdelity

simulation results.

2.2.1 The graph auto encoder architecture169

We adopt the graph isomorphism network (GIN) (cf. Xu et al. [61]) to perform the embedding task. We170

use GIN because it is a message-passing model capable of discriminating different graph structures iden-171

tiﬁed by the Weisfeiler-Lehma isomorphism test [60]. Providing that we use a proper graph pooling layer,172

the embedding of GIN is inherently permutation invariance, which means that the ordering of the nodes173

will not affect the predictions. More importantly, the fact that GIN passes the isomorphism test enables us174

to distinguish non-isomorphic subgraphs by mapping them onto different encoded latent vectors and vice175

versa – a feat that cannot be achieved by the conventional graph convolutional network and GraphSAGE176

[61]. These two features combined improve the expressive power of the GIN such that the relationships177

among ﬁnite element nodes can be captured by the neural network. Fig. 3shows the architecture of the178

graph autoencoder designed for the shell problems. The encoder part of this architecture takes in the ad-179

jacency matrix and feature matrix of the input graph G0

in :(Ain,Xin )and produces an encoded feature180

vector henc, which is denoted as a functional expression: henc =Enc(Xin ,Ain). The decoder part of this ar-181

chitecture then utilizes the encoder output, together with the adjacency matrix of the output graph Aout for182

GIN convolution, to produce a decoded feature matrix ˜

Xthat completes an output graph G0

out :(Aout,˜

X),183

which could be written as ˜

X=Dec(henc,Aout ). Problem formulation is generally based on supervision184

of the decoded output ˜

X, which will be the focus of the following sections. The rest of this section will185

present details about how the layer components shown in Fig. 3operate.186

We ﬁrst introduce one of the most widely-used architectures called multi-layer perceptron (MLP),187

which is included as a substructure in our graph autoencoder architecture. MLP is a functional approx-188

imation expressed as follows:189

MLP(X) = W(K)·act(W(K−1)·act(...act(W(1)·X+b(1))...) + b(K−1)) + b(K)(1)

where Wand bis called the weight matrix and the bias vector of the MLP subtructure, respectively. the190

superscript (K)indicates the K-th layer of the MLP substructure. act(·)is called the activation function of191

individual layers; here we adopt the rectiﬁed linear unit (ReLU) for act(·)such that ReLU(x) = xif x>0192

otherwise ReLU(x) = 0.193

We then focus on the GIN convolution layers, which take in some adjacency matrix and feature matrix194

and output an embedded feature matrix. The matrix formulation of a GIN layer is:195

H(k)=MLP(k)(A+ (1+e)I)·H(k−1)(2)

6 Mian Xiao et al.

Fig. 3: Graph autoencoder architecture. Notice that the input and output graphs are not necessarily sharing

the same adjacency matrix.

where the superscript (k)indicates the k-th layer in the architecture; His the embedded nodal feature196

matrix coming from the output of the previous layer with H(1)=Xat the input layer. eis a learnable197

parameter. For consecutive GIN convolution layers, the following layer accepts the same Aas the previous198

layer. For the beginning GIN layer in both the encoder and the decoder, Ashould be prescribed as either199

Ain or Aout.200

Our architecture also includes global operations on the graph. The graph global mean pooling opera-201

tion in the encoder performs the following computation:202

havg =1

∑

i=1

hi(3)

where hiis the embedded nodal feature of vicorresponding to the i-th row of H, and havg is the resultant203

graph feature vector from global mean pooling.204

The broadcasting operation in the decoder is in fact a matrix reshape operation that converts a row205

vector hdec of size NDenc coming from the output of an MLP substructure in the decoder, to an embedded206

nodal feature matrix of size N×Denc such that:207

hij =(hdec )j+(i−1)Denc (4)

where hij indicates the j-th component of the embedded nodal feature vector hiafter broadcasting.208

2.2.2 Finding the low-dimensional latent space of the high ﬁdelity shell response209

In the previous section, we discuss the strategy to construct the mapping F(lG0

j) = h˜

jsuch that h˜

j∼210

lG0

jto established an augmented data set h˜

G. We assume that Fis learned in a supervised manner: we211

aim to minimize the discrepancy between the nodal feature matrix of training labels and the approximated212

nodal feature matrix output by the neural network. We intuitively construct the training labels with hG,213

and thus we required that for each snapshot in hGthere exist some snapshot in lGcorresponding to the214

results with the same loading condition, which is summarized as follows:215

∃lh G⊂lGs.t. hG=nF(lh G0

j)|lh G0

j∈lh Go(5)

Geometric learning for shells 7

The subset lh Gthen constructs the training input set. We next approximate Fwith the graph autoen-216

coder proposed in Section 2.2.1 mainly computing the decoded nodal features as ˜

X=ˆ

F(X). The loss217

function is adopted as the node-wise mean square error of the nodal feature matrix, which leads to the218

following training objective:219

min

ΘF

Nlh

∑

i=1



ˆ

F(Xlh

(i))−Xh

(i)



fro ,ˆ

F(X) = DecF(EncF(X,Al),Ah)(6)

where ΘFis the collection of all trainable network parameters of ˆ

F(·).Nlh

sis the size of lh Gas well as hG.220

The subscript (i)indicates the i-th sample in hGor lh G, while the sample sequence satisﬁes hG0

i∼lh G0

221

with hG0

i:(Ah,Xh

(i))and lh G0

i:(Al,Xlh

(i)). The operator k·kfro indicates the Frobenius norm of a matrix.222

The subscript Fof Dec and Enc indicates the decoder and encoder function for the graph autoencoder223

approximating F, in order to differentiate from the reconstruction autoencoder mentioned in the following224

part. The approximated mapping ˆ

F(·)helps us to enrich the high ﬁdelity dataset as follows:225

h˜

G=





h˜

j:(Ah,˜

(j))|˜

(j)=





F(Xl

(j)),lG0

j:(Al,Xl

(j))∈lG\lh G,

(j),lG0

j:(Al,Xl

(j))∈lh G.





(7)

Here we realize the signiﬁcance of having a low-ﬁdelity dataset lGand the mapping F: we are now226

able to construct a relatively large high-ﬁdelity dataset to improve the reduced ordered model without227

spending too much effort performing experiments on the high ﬁdelity scales.228

With h˜

Gpopulated from lG, we are ready to formulate the graph embedding problem that determines229

the reduced ordered latent space. The general idea is to construct a graph autoencoder function ˆ

R(·)whose230

output approximates its input itself. We still adopt the loss function as the node-wise mean square error for231

the feature matrix between the labels in h˜

Gand the output from ˆ

R(·), which yields the following training232

objective:233

min

ΘR

Nhh

∑

i=1



ˆ

R(˜

(i))−˜

(i)



fro ,ˆ

R(X) = DecR(EncR(X,Ah),Ah)(8)

where ΘRis the collection of all trainable network parameters of ˆ

R(·).Nhh

sis the size of h˜

Gas well as234

lG. The subscript Rindicates the encoder and decoder function of the reconstruction autoencoder. The235

reduced ordered latent space Lis then deﬁned as the space spanned by henc =EncR(X,Ah)for arbitrary236

Xcoming from an admissible deformed shell conﬁguration G0, where L⊂RDenc .237

2.2.3 Predicting the high-ﬁdelity results without full-scale simulations238

This section presents how we utilize the parametrized load pto predict the actual deformed shell con-239

ﬁguration G0. As we ﬁnd the reduced ordered latent space Lin the previous section, we may notice that240

Lis generally entangled, which does not ideally captures the reduced ordered dynamics in the maximum241

sense. We here propose to construct a governing law on Lbased on pcorresponding to the shell con-242

ﬁguration of interest to disentangle the reduced ordered latent space, denoted as a functional expression243

henc =f(p). We then approximately parametrize the governing law function as ˆ

f(·)by a simple feed-244

forward neural network with MLP architecture. We ﬁt ˆ

f(·)with the mean square error loss between the245

encoded feature vector obtained from the graph autoencoder and that computed by ˆ

f(·), which yields the246

following learning objective:247

min

Θf

Nhh

∑

i=1



ˆ

f(p(i))−EncR(˜

(i),Ah)



2(9)

where Θfis the collection of all trainable network parameters of ˆ

f(·).p(i)is the loading condition corre-248

sponding to the i-th snapshot h˜

i:(Ah,˜

(i)). The operator k · k2is the vector Euclidean norm.249

8 Mian Xiao et al.

After we learn the neural network approximation of f(·), the prediction of high ﬁdelity results is to250

determine the nodal feature matrix ¯

Xgiven some loading condition ¯pas follows:251

X=DecR(ˆ

f(¯p),Ah)(10)

In essence, we introduce a graph neural network approach to construct a response surface. Since (1)252

the data are obtained from the simulations that obey the balance principles and (2) a successful embedding253

should be capable of preserving the relationships among nodes, we hypothesize that this will give us more254

accurate and robust predictions than other alternatives that employ basis functions that directly interpolate255

the hypersurface in the ambient space RN+P. This hypothesis will be tested in Section 5. For the domain256

of interest where the balance principle must be precisely enforced, we introduce a locally-enhancement257

technique that can improve the ﬁdelity via the tangent space of the nonlinear solution manifold.258

3 Local enrichment with geometrical and physical constraints on the machine learning prediction259

The previous section shows how the graph representation of a deformed shell manifold is predicted.260

However, the point cloud obtained with (10) may exhibit undesirable high-frequency local oscillation if the261

training data is limited [11]. This non-physical oscillation due to data compression is undesirable but quite262

commonly found in neural network predictions if the training data set is limited and the proper regular-263

ization is not used [11,57]. Since the change of the curvature of a shell under stable deformation requires264

the change of elastic stored energy, the curvature of the learned solution can be, in principle, regularized265

via an energy functional. In the ﬁeld of graph-manifold learning, techniques that control the smoothness266

and curvature has evolved from graph-Laplacian-based (spectral graph theory) methods [14,23,63], to267

curvature ﬁlters that are directly derived from Riemannian geometry [19,21,53,64]. Curvature ﬁlters are268

shown to be minimizing variational energy [18], which justiﬁes its usage as a geometry optimization tool269

and the potential compatibility with the physical governing equations.270

The physical balance laws constitute another important type of constraint in the veriﬁcation of ma-271

chine learning predictions for mechanics problems. They can be enforced with a regularization term in272

the training objective [3,4,57], where a notable example is the physics informed learning [46], or with prior273

assumptions in the solution functions [55,62], where a notable example is the neural operator [35].274

As a result, we propose to enforce constraints on the local curvature of the predicted point cloud ¯

X. In275

the meantime, we reparametrize a local region of interest on the shell conﬁguration as a linear combination276

of functional basis derived from the direct neural network prediction. We ﬁnally determine the correction277

of neural network predictions from Section 2by computing physical properties (e.g. value of the residual278

equation) and formulate the optimization problem accordingly. The rest of this section is organized as279

follows: Section 3.1 presents the formulation of a surface (locally) and the governing equations of the shell280

problem; Section 3.3 will provide numerical implementation details of these constraints.281

3.1 Ingradients of the geometrical and physical constraints282

At the beginning of this section, we show how to express a surface mathematically to approximate the283

deformed shell conﬁguration, as a foundation of curvature deﬁnition. We are speciﬁcally interested in the284

curvature properties of 2nd order, like Gaussian curvature, which is associated with the ﬁrst and second285

fundamental form of a surface [52]. Thus, we would like to describe the mathematical expression of the286

ﬁrst/second fundamental form and Gaussian curvature in the following few paragraphs. For convenience,287

we denote the actual spatial coordinates of a point on the shell as (x1,x2,x3)and a local coordinate system288

of this point originating at (x1,x2,x3)as (y1,y2,y3), where the local coordinate system is orthonormal289

with plane y1−y2corresponding to the local tangential plane and y3direction correspond to the normal290

direction.291

Geometric learning for shells 9

3.1.1 Gaussian curvature292

We assume that the local surface equation can be mathematically expressed as: y3−ω(y1,y2) = 0,293

which leads to a vectorial form of the surface coordinates z= (y1,y2,ω(y1,y2)) parametrized with solely294

y1and y2. The ﬁrst fundamental form is the inner product of two tangent vectors yt

1,yt

2at some speciﬁc295

location:296

I(yt

1,yt

2) = yt

1·yt

2=hα1α2i"g11 g12

g12 g22#"α3

α4#where (yt

1=α1∂z

∂y1+α2∂z

∂y2

2=α3∂z

∂y1+α4∂z

∂y2

(11)

where I(·,·)is the ﬁrst fundamental form operator. "g11 g12

g12 g22#is called the matrix form of I(·,·), which is297

also the matrix form of the metric tensor g. Its components are computed as follows:298

g11 =∂z

∂y1

·∂z

∂y1

=∂ω

∂y1

∂ω

∂y1

,g12 =∂z

∂y1

·∂z

∂y2

=∂ω

∂y1

∂ω

∂y2

,g22 =∂z

∂y2

·∂z

∂y2

=∂ω

∂y2

∂ω

∂y2

(12)

For some speciﬁc location zon the surface of interest, the second fundamental form is deﬁned as a299

function of a local perturbation in the parameteric coordinates (∆y1,∆y2):300

II(∆y1,∆y2) = h∆y1∆y2i"b11 b12

b12 b22#"∆y1

∆y2#(13)

where II(·,·)is the second fundamental form operator. The matrix components in (13) are computed as301

follows:302

b11 =∂2ω

∂y2

,b12 =∂2ω

∂y1∂y2

,b22 =∂2ω

∂y2

(14)

The Gaussian curvature Kis then deﬁned as follows:303

K=det(I I)

det(I)=b11b22 −b2

g11g22 −g2

(15)

where det(II), det(I)is the determinant of the matrix form of the second fundamental form and the ﬁrst304

fundamental form, respectively.305

3.1.2 Balance law constraints306

We next focus on presenting a brief summary of the geometrically exact shell model we adopt [49], for307

the convenience of introducing the governing equations as physical constraints. In this model, we consider308

the shell conﬁguration as a 2D Riemann manifold with ﬁnite extent in the actual 3D space, where the309

positional information of the shell is indicated by the mid-surface location ϕ. The geometrically exact310

feature is achieved by introducing a vector ﬁeld t, also called director, to account for the bending and shear311

effect of the shell. These vector ﬁelds ϕand tformulate the basic representation of the shell geometry312

and facilitate the deﬁnition of shell strain measures and the evaluation of shell stress measures. Due to313

this two-ﬁeld mixed formulation, there exist two residual equations as the target of the physical constraint314

evaluation: one is associated with linear momentum balance, denoted as Rn; the other is associated with315

director momentum balance, denoted as Rm. Necessary ingredients for evaluating physical constraints are316

summarized in Table 1. Meanwhile, the shell response is assumed quasi-static.317

Remark 2 ϕand tare deformable mappings from a static 2D parametric space A ⊂ R2to the actual 3D318

space R3and the unit sphere S2. For notational convenience, we use (ξ,η)to denote the parametric co-319

ordinates in Afrom now on. This enables us to express the shell conﬁguration mappings in functional320

forms:321

ϕ(ξ,η) = [ϕ1(ξ,η)ϕ2(ξ,η)ϕ3(ξ,η)]T(16)

10 Mian Xiao et al.

and322

t(ξ,η) = [t1(ξ,η)t2(ξ,η)t3(ξ,η)]T, (17)

where ϕiand ti(i=1, 2, 3) indicate the componentwise location and director ﬁelds, respectively. All super323

and subscripts α,β,µlisted in Table 1are associated with these parametric coordinates implicitly, where324

the number 1 corresponds to ξand the number 2 corresponds to η.325

governing equations Nomenclature or explanation

surface frame

and jacobian

a0α=ϕ0,α,aα=ϕ,αa0α,aα

initial and convected surface frame

vector (tangential basis)

J=ka1×a2k2Jsurface jacobian

kinematic relationships

aαβ =aα·aβ

γα=aα·t

καβ =aα·t,β

aαβ

γαβ

καβ

kinematic variables for computing

the effective strain measures

in the following equations

effective strains

εαβ =1

2(aαβ −a0αβ )

δα=γα−γ0α

ραβ =καβ −κ0α β

εαβ effective membrane strain

δαeffective shear strain

ραβ effective coupled strain

frame vector constraint

t,α=λµ

αaµ+λ3

αt

t·t,α=0

λ3

α=−λµ

αγµ

λµ

coefﬁcients of the frame vectors

aµ,tafter the

vector decomposition of t,α

linear momentum balance Rn=1

J(Jnα),α=0nαstress resultant

director momentum bal-

ance

Rm=1

J(J˜mα),α−¯

l=0 ˜mα,¯

lstress couple, across-the-thickness

stress resultant

decomposition of stress

resultants and stress couples

nα=nβαaβ+qαt

˜mα=˜

mβαaβ+˜

m3αt

l=¯

λt+ ( ˜

qα+λα

µ˜

m3µ)aα

nβα,qα

mβα,˜

m3α

stress resultant components

effective stress couple components

Lagrange multiplier of t

effective stress resultants ˜

nβα =nβα −λβ

µ˜

mαµ

qα=qα+λβ

µγβ˜

mαµ

nβα

qα

effective membrane stress

effective shear stress

Constitutive laws

ψ(εβα,δα,ρβα)

nβα =∂ψ/∂εβα

qα=∂ψ/∂δα

mβα =∂ψ/∂ρβα

ψassumed elastic energy functional

Table 1: Summary of the governing equations of the quasi-static geometrically exact shell model adopted

for verifying the physical constraints locally. All super and subscripts α,β,µ=1, 2.

3.2 Curvature ﬁlter326

To enforce the geometric constraints and physical constraints in our machine learning model, a straight-327

forward approach is to incorporate these constraints as additional terms into the training loss function [57].328

One improtant issue we should notice is that in the computation of local curvature, it is required to com-329

plete an eigenvalue decomposition (in normal vector estimation) and a local regression problem by solving330

linear equation systems. The corresponding loss term are expected to be in an extremely complicated and331

highly nonlinear form, which brings great challenges to the gradient computation with respect to the train-332

able parameters. Thus the training progress is greatly prolongated if geometrical and physical constraints333

are enforced concurrently. For efﬁciency, we decide to apply the geometric constraints and physical con-334

straints as a post-processing step after we trained the model that predicts the discrete ﬁnite element graph.335

Geometric learning for shells 11

The rest of this section will brieﬂy demonstrate the application geometric constraints (curvature control).336

For the physical constraints application, we refer to Section 3.3.337

We adopt the Gaussian curvature ﬁlter proposed in [53] to smooth the locally oscillatory surface predic-338

tion. The basic idea of Gaussian curvature ﬁlter is to develop an algorithm to adjust the discrete geometry339

such that the total Gaussian curvature energy RΩKdΩover the domain of interest (Ω) is minimized. Ac-340

cording to derivation in 3.1.1, the local Gaussian curvature Kis computed based on the gradients evaluated341

in the tagent space, and Tang et al [53] further show that it can be approximated by the relative coordinates342

of one-hop neighbors for a node in a discrete mesh. Using a gradient descent approach, it can be derived343

that in each iteration for minimizing the curvature energy, the coordinate adjustment for each node is344

function of its one-hop neighbors’ relative coordinates. This generalizes to a simple but efﬁcient algorithm345

framework summarized in Algorithm 1346

Algorithm 1 Improve model prediction with Gaussian curvature ﬁlter

1: while Convergence not achieved do

2: Loop over all nodes:

3: For each node, collect its node coordinate and all the one-hop neighbors’ coordinate, then evaluate

the direction to update node coordinate ˜qand the magnitude ˜

∆(for mathematic details, cf [53]).

4: Update the node coordinate: x=x+˜

∆˜q.

3.3 Reduced-order modeling via tangential space of the solution manifold347

In the previous section, we argue that the shell conﬁguration vector ﬁelds ϕand tprovide sufﬁcient in-348

formation for evaluating the geometric and physical constraint equations. One possible remedy to improve349

the compatibility of the balance principles without sufﬁciently slowing down the prediction is to introduce350

Galerkin projection via a collection of orthogonal bases. In theory, these orthogonal bases can be generated351

through principal component analysis performed on the entire collection of solutions, and the ﬁdelity can352

be improved by incorporating more orthogonal bases at the expense of computational speed. However,353

this global linear embedding strategy is known to yield an unsatisfactory ﬁdelity-speed tradeoff for highly354

nonlinear problems. As such, we introduce an empirical approach in which we establish a locally linear355

embedding where we identify the tangential space at the parametric load pwhere we sought the solution.356

Then we use the basis of this tangential space to establish the Galerkin projection such that the resultant357

Gakerlin projection may vary according to the types of loadings p, a technique commonly used to model358

high-dimensional data (see, for instance, Donoho and Grimes [12]). A similar strategy has also been used359

in He and Chen [20], He et al. [22] where a locally convex space is established via points in a constitutive360

manifold for data-driven simulations.361

As a result, we may formulate the problem for enforcing physical constraints as an optimization prob-362

lem to minimize the norm of residual vectors Rn,Rmwith ϕand tas governing variables. Functional363

variables ϕand tare parametrized as a linear combination of functional basis that is perturbed from the364

predicted solution:365











ϕ=c0ˆϕ|p+∑˜

kckˆϕ|p+δpk

t=c0ˆ

t|p+∑˜

kckˆ

t|p+δpk

1=c0+∑˜

kck

(18)

where ˜

Kis the number of functional basis. ˆϕand ˆ

tare trained neural network functions for the local366

solution predicted at the load speciﬁed by the load vector in the subscript (p+δpk), and non-degenerated367

basis is preferred such that: Rank(span{ˆϕ|p+δp1,· · · ˆϕ|p+δp˜

K}) = ˜

K, Rank(span{ˆϕ|t+δt1,· · · ˆϕ|t+δt˜

K}) =368

K. With this parametrization of the target function, we can formulate the physics-based solution adjustment369

as an optimization problem with coefﬁcients ckas governing variables instead of the inﬁnite-dimensional370

12 Mian Xiao et al.

functional space:371

min

c0,c1,··· ,c˜

kRnk2+˜

wk[t,β·Rm]k2s.t. c0+

∑

ck=1 (19)

where ˜

wis a tunable factor that controls the relative importance of the director momentum residual in372

the minimization progress with respect to the linear momentum residual. Instead of adopting Rmin the373

optimization objective, we use a 2D vector [t,β·Rm](β=1, 2)for the consideration of director momentum374

equation, which will be explained in the rest of this section.375

Algorithm 2 Improve model predictions with physical constraints around a neighborhood of one ﬁnite

element node ¯

1: Prescribe δp1,· · · ,δp˜

Kand predict the deformed shell conﬁgurations at the ﬁve different loading con-

ditions p,p+δp1,· · · ,p+δp˜

Kwith (10). ˜

Kis generally selected according to the dimension of the

loading parametric space.

2: Find a neighborhood nodes set of ¯

v0, and extract the local ˆϕand ˆ

tﬁeld based on the neighborhood

node set for the ˜

K+1 loading conditions in step 1.

3: Train ﬁve different MLP functions to approximate the perturbed local solutions. Notice that ﬁtting ﬁve

components are sufﬁcient: ϕ1,ϕ2,ϕ3,t1,t2, since we can determine t3by t3=±q1−t2

1−t2

2where the

sign is up to the orientation of the shell conﬁguration.

4: Establish a nonlinear optimizer to solve the problem described by (19).

We next focus on deriving the balance laws listed in Table 1and implementing them based on the376

locally approximated manifold expressions. We notice that the variables ˜

m3α,¯

λcannot be derived from377

either kinematic relationships or constitutive relationships. In this sense, we aim to clear out the terms378

with ˜

m3α,¯

λin the expanded expression of Rm. We accomplish it by taking the dot product of Rmwith379

respect to t,β, where we ﬁrst expand Rmas follows:380

Rm= ( ˜

mµα,αaµ+˜

m3α,αt+˜

mµαaµ,α+˜

m3αt,α) + J,α

J(˜

mµαaµ+˜

m3αt)−¯

λt−(˜

qα+λα

µ˜

m3µ)aα(20)

We then compute t,β·Rmand simplify it with t,β·t=0 and t,β·aα=καβ :381

t,β·Rm= ( ˜

mµα,ακµβ +˜

mαµt,β·aµ,α+˜

m3αt,β·t,α) + J,α

J˜

mµακµβ −(˜

qα+λα

µ˜

m3µ)καβ (21)

The frame vector decomposition expression in Table 1shows us t,α·t,β=λµ

αaµ·t,β+λ3

αt·t,β=λµ

ακµβ .382

This concludes the equivalence between two terms in (21): ˜

m3αt,β·t,α=λα

µ˜

m3µκαβ . As a result, the director-383

gradient-weighted director momentum residual in (21) is ﬁnnaly computed as follows:384

t,β·Rm=˜

mµα,ακµβ +˜

mµαt,β·aµ,α+J,α

J˜

mµακµβ −˜

qακαβ (22)

In contrast to Rm,Rnpossesses an expanded expression consisting of ˜

nαβ ,˜

qα,˜

mαβ ,λβ

µ,γβ,aα,tand385

some of their gradient or divergence with respect to the parametric coordinates. These variables can be386

computed via the constitutive/kinematic equations or other geometric constraints listed in Table 1. Hence387

we no longer perform further derivation on Rnfor numerical implementation purposes. The procedures388

for computing the residuals of the linear and director momentum balance are then summarized in Algo-389

rithm 3:390

Geometric learning for shells 13

Algorithm 3 Compute the residual around a neighborhood of one ﬁnite element node ¯

Given:ϕand taround ¯

v0in the form of (18) with coefﬁcients c0· · · c4determined.

1: Compute the surface frame vectors a0α,aαand the surface jacobian J.

2: Compute the t,αwith the director ﬁeld obtained in Step 2.

3: Compute the kinematic variables aαβ ,γα,καβ in both the current and the initial conﬁguration.

4: Compute effective strain measures εαβ,δα,ρα β.

5: Find the coefﬁcients λµ

αfrom vector decomposition of t,α.

6: Evaluate the effective stress measures ˜

nαβ ,˜

qα,˜

mαβ with the constitutive equations (ref constitutive eqn)

7: Compute Rn=1

J(Jnα),αand t,β·Rmwith (22).

4 Training of the geometric leanring model391

In this section, we show two numerical examples that demonstrate the correctness and applicability of392

our model, and the procedures to generate the dataset with different levels of ﬁdelity via ﬁnite element393

simulation. In Section 4.1, we present the problem conﬁgurations of the two examples of interest, with394

details about the shell geometry, material properties, and loading conditions. In Section 4.2, we introduce395

mesh discretizations with multiple resolutions and the simulation setup to demonstrate how data are sam-396

pled within load parametric space. We additionally list the hyperparameters assumed at the model training397

phase in Section 4.3.398

4.1 Shell problem conﬁgurations399

4.1.1 A hemispherical shell holed at the pole400

The shell problem conﬁguration presented in this subsection is in a hemispherical shape but with a401

circular hole punched at the pole. The edge of the hole coincides with a latitude line at a zenith angle of402

18◦. According to the symmetry of this conﬁguration, we take one-quarter of the holed hemisphere as the403

computation domain. The initial shell conﬁguration can be expressed as follows, assuming that the initial404

director ﬁeld matches the outward normal vector ﬁeld of the shell surface:405











ϕ1(ξ,η) = ¯

rcos(π

2ξ)cos(2π

5η),

ϕ2(ξ,η) = ¯

rsin(π

2ξ)cos(2π

5η),

ϕ3(ξ,η) = ¯

rsin(2π

5η).











t1(ξ,η) = ϕ1(ξ,η)/¯

t2(ξ,η) = ϕ2(ξ,η)/¯

t3(ξ,η) = ϕ3(ξ,η)/¯

0≤ξ,η≤1 (23)

where ¯

ris the spherical radius chosen to be 10 m in our case.406

The domain symmetry requires symmetric boundary conditions on the two lateral bounds located at407

ξ=0 and ξ=1. The driving loads are prescribed as concentrated force at two locations: at ξ=0, η=0408

it points toward the positive spatial xaxis, while at ξ=1, η=0 it points toward the negative spatial y409

axis. The two forces possess the same magnitude, indicating that there is just one parameter ¯

Fcontrolling410

the shell dynamics. The material properties are as follows: Young’s modulus is 68.25 MPa, and the Poisson411

ratio is 0.3. The thickness of this structure is assumed to be 0.04 m. We plot the shell conﬁguration and the412

loading conditions (projected to xy plane) in Fig 4.413

4.1.2 A doubly-curved shell anchored at four corners414

The second problem simulates a doubly-curved shell structure anchored at four corners for shelter415

purposes. The geometry of this shell conﬁguration can be generalized with the following equation: z−416

ω(x,y) = 0, where ω(x,y)is expressed as follows (unit is meter for x,y,z):417

ω(x,y) = 2.4(1−ω1(x,y)2) + ω3(x,y)2h3(1−ω2(x,y)2)−2.4(1−ω1(x,y)2)i,−1≤x,y≤1 (24)

14 Mian Xiao et al.

(a) shell domain (b) boundary conditions (pro-

jected to xy plane)

Fig. 4: Computational domain and boundary conditions of the initial conﬁguration of the hemispherical

shell holed at the pole.

where418

ω1(x,y) = |x|+|y|

2,ω2(x,y) = |x+y|−|x−y|

2,ω3(x,y) = |x+y|+|x−y|

2(25)

We present the spatial view of the initial conﬁguration from one perspective and plot the zelevation as419

a function of x,yin Fig 5. We still assume that the initial director ﬁeld corresponds to the outward normal420

vector ﬁeld of the shell surface. As this shell is anchored at four corners, we enforce ﬁx boundary conditions421

at the four corners. The driven force is concentrated and located at (x,y,z) = (0, −1, 3)and (x,y,z) =422

(0, 1, 3), where the forces have only xand ycomponent. In total, the external load can be represented with423

four parameters. The material properties are as follows: Young’s modulus is 29 MPa, and the Poisson ratio424

is 0.3. The thickness of this structure is assumed to be 0.04 m.

(a) shell conﬁguration

in one perspective

(b) zdirection elevation plot in xy plane (c) boundary conditions (pro-

jected to xy plane)

Fig. 5: Geometry of the initial conﬁguration of the doubly-curved shell anchored at four corners together

with the boundary conditions. x,y,zhave the same length unit as meter. Triangulars in (c) indicates ﬁx

boundary conditions (in all directions) at a speciﬁc point.

425

4.2 Generation of the graph database for deformed shell ﬁelds426

In this section, we focus on how simulations are performed to collect snapshots of deformed shell427

conﬁgurations. For the sake of dataset enrichment mentioned in Section 2.1, we discretize the conﬁgura-428

tions of both numerical examples into ﬁnite element meshes with different resolutions. In this sense, the429

Geometric learning for shells 15

low-ﬁdelity dataset lGis established by collecting simulation snapshots in the coarser mesh, while the430

high-ﬁdelity dataset hGis established with simulation snapshots in the ﬁner mesh. We usually simulate431

fewer steps (snapshots) in the ﬁner mesh than that in the coarser mesh.432

During mesh discretization, we ﬁrst mesh the parametric domain to obtain a set of node coordinates (in433

2D) and a set of element connections. We then perform a coordinate transformation that maps the paramet-434

ric coordinate to its spatial counterpart on the shell manifold. We ﬁnally deﬁne the 3D mesh conﬁguration435

by combining the transformed node coordinates and the same element connections as those in the 2D436

parametric mesh. Fig. 6shows the ﬁnite element meshes in different resolutions for the ﬁrst example: a437

hemispherical shell holed at the pole. Fig. 6(a,c) demonstrates the 3D shell mesh conﬁguration from some438

perspective, while Fig. 6(b,d) shows the mesh discretization in the 2D parametric space before coordinate439

transformation. The coarser mesh (Fig. 6(a,b)) has 125 nodes and 107 elements, while the ﬁner mesh (Fig.440

6(c,d)) has 1781 nodes and 1712 elements.441

(a) (b) (c) (d)

Fig. 6: Mesh discretization of the holed hemispherical shell. (a), (b) are the ﬁnite element mesh with coarse

resolution viewed in the actual 3D space and the parametric space, respectively. (c), (d) are the ﬁnite ele-

ment mesh with ﬁne resolution viewed in the actual 3D space and the parametric space, respectively.

We simulate 200 uniform-increment steps with the coarser mesh to build lG, while the loading force442

value ¯

Fgrows linearly to 1. To generate hGwith the ﬁner mesh (Fig. 6(c)), we maintain the ¯

Fvalue in443

the last step as the same ultimate load magnitude in the simulation setup with the coarser mesh (equal444

to 1) and load it for 20 uniform-increment steps. In this sense, for the 10th, 20th ,..., 200th snapshots in445

lGif counted sequentially in time, there is a snapshot in hGcorresponding to the same prescribed load446

F.lh Gis then established by extracting the last snapshot of every ten steps lGsorted by time. We also447

simulate additional steps with the ﬁner mesh to build a dataset for testing prediction accuracy, where we448

prescribe the external force as 0.6275 and 0.9425 such that there are no snapshots in the training dataset449

that correspond to these loads.450

Fig. 7presents the ﬁnite element meshes in different resolutions for the second example: a doubly451

curved shell anchored at four corners. The coarser mesh (Fig. 7(a,b)) has 397 nodes and 356 elements,452

while the ﬁner mesh (Fig. 7(c,d)) has 1521 nodes and 1424 elements. To prescribe the load, we randomly453

sample 400 ultimate load values and prescribe a monotonic straight-line loading path to each ultimate load454

point. On each loading path, we simulate 20 steps. Fig. 8demonstrates how the 400 loading paths populate455

the load parameter space:456

To construct lGfor this case, we simulate all steps on each loading path with the coarser mesh (Fig.457

7(a,b)), which yields in total 8,000 snapshots. When establishing hG, we randomly sample 40 load paths458

within the paths provided in Fig. and simulate them with the ﬁner mesh (Fig. 7(c,d)), which yields in total459

800 snapshots. For testing purposes, we prescribed three test loading paths: one is seen in lG\lhGwhile460

the other two is unseen in training set as Fig. 9shows.461

Remark 3 List the simulation time and show how we save time by simulating on multiple resolutions. If462

we do not establish a low-ﬁdelity dataset for data enrichment, we should simulate on the ﬁner mesh for as463

many steps as those on the coarser mesh, in order to maintain the same data density. The time saved when464

16 Mian Xiao et al.

(a) (b) (c) (d)

Fig. 7: Mesh discretization of the doubly curved shell anchored at four corners.(a), (b) are the ﬁnite element

mesh with coarse resolution viewed in the actual 3D space and the parametric space, respectively. (c), (d)

are the ﬁnite element mesh with ﬁne resolution viewed in the actual 3D space and the parametric space,

respectively.

(a) Loading paths projected to ¯

F0x¯

F0y

plane.

(b) Loading paths projected to ¯

F1x¯

F1y

plane.

Fig. 8: Prescribed loading paths to generate training dataset for the doubly-curved shell example. All 400

paths are projected to either ¯

F0x¯

F0yplane or ¯

F1x¯

F1yplane as straight lines, and the ultimate load at the end

of each path is marked by a circle. (as Fig. 5(c) shows, ¯

F0x,¯

F0yis the force applied at (0, −1)and ¯

F1x,¯

F1yis

the force applied at (0, 1)) The overbar is omitted in the axes label texts for simplicity.

resolution steps per path avg. time per path total sim. time

hemispherical shell coarse 200 23 s 23 s

ﬁne 20 59 s 59 s

doubly curved shell coarse 20 11 s 4,400 s

ﬁne 20 68 s 2,720 s

Table 2: Simulation time of the two examples at different resolutions when generating the training dataset.

generating the training data with low-ﬁdelity data enrichment is: 59 ×10 −(23 +59) = 508 s (for the ﬁrst465

example) and 68 ×400 −(4400 +2720) = 20, 080 s (for the second example)!466

4.3 Training conﬁguration of the geometric models467

This section aims to demonstrate the training conﬁguration for the geometric learning model of both468

numerical examples, with the majority of focus on hyperparameters. For each example, we would like to469

Geometric learning for shells 17

(a) Test loading paths projected to ¯

F0x¯

F0y

plane.

(b) Test loading paths projected to

F1x¯

F1yplane.

Fig. 9: Loading paths for modeling testing for the doubly-curved shell example. The overbar is omitted in

the axes label texts for simplicity.

sequentially introduce the training setup for the mapping ˆ

F, the graph autoencoder ˆ

R, and the shell gov-470

erning law ˆ

f. Training setups usually cover the dimensions of neural network layers or substructures, the471

choice of optimizer, the training iterations, and the learning rate. The neural networking is trained using472

the PyTorch open source library [44] on an NVIDIA DGX A100 GPU workstation. The training conﬁgu-473

ration for both examples is summarized in Table 3and 4. We would also like to demonstrate the success474

of this training setup in Fig. 10 with the recorded loss history trained and validated on the training and475

testing dataset for the doubly-curved shell example.

Ftraining setup

features of the GIN layers encoded dimensions 8

latent channels 64

MLP subtructure of GIN layers num. of layers 3

hidden neurons 100

features of the optimizer

batch size 1

training iterations 400

learning rate 5 ×10−4

Rtraining setup all setup are the same as those while training ˆ

Fexcept that batch size is 16

ftraining setup

MLP architecture num. of layers 3

hidden neurons 100

features of the optimizer

batch size 16

training iterations 500

learning rate 5 ×10−4

Table 3: Training setup of the ﬁrst example (heimispherical shell). We omit the optimizer type as we adopt

the Adam optimizer [1] for all tasks.

476

5 Numerical results and discussion477

This section follows the demonstration in Section 4and presents modeling results of the two examples478

of interest. Section 5.1 shows veriﬁcation results on the holed spherical shell to justify the usage of cur-479

vature ﬁlters and the correctness of the resulting prediction. Section 5.2 shows miscellaneous results that480

18 Mian Xiao et al.

Ftraining setup

features of the GIN layers encoded dimensions 16

latent channels 64

MLP subtructure of GIN layers same as Table 3

features of the optimizer

batch size 32

training iterations 500

learning rate 2.5 ×10−4

Rtraining setup

features of the GIN layers same as those in ˆ

Ftraining setup

MLP subtructure of GIN layers same as those in ˆ

Ftraining setup

features of the optimizer same as those in ˆ

Ftraining setup

except that training iteration is 250

ftraining setup

MLP architecture same as Table 3

features of the optimizer

batch size 128

training iterations 1000

learning rate 5 ×10−4

Table 4: Training setup of the second example (doubly-curved shell). We omit the optimizer type as we

adopt the Adam optimizer for all tasks.

(a) ˆ

Ftraining history (b) ˆ

Rtraining history (c) ˆ

ftraining history

Fig. 10: Training and validation loss function values for different neural network training tasks listed in

this section (for the doubly-curved shell example).

demonstrate the capacity of our model in reconstructing a realistic response surface in higher-dimensional481

parametric space.482

5.1 Spherical shell veriﬁcation483

The ﬁrst example mainly serves as a veriﬁcation example showing that the response surface can be484

successfully reconstructed in a lower dimensional space. As the parametric loading space is only one-485

dimensional, we select two loading states whose simulated shell conﬁguration is not included in the train-486

ing dataset: ¯

F=0.6275, 0.9425. We plot the predicted shell manifold against the conﬁguration obtained487

from direct numerical simulations (DNS) in Fig. 11, where we can easily observe the predicted conﬁgu-488

ration is very close to the reference one (from DNS). Physical residual equation values are also evaluated489

within a neighborhood of one of the loading points ((ξ,η)=(1, 0)), as presented in Fig. 12, which sug-490

gests that the predicted shell conﬁguration is in fact well compatible with the physical constraints equation491

within the local area around where the concentrated load applied. Notice that the value of the residuals are492

normalized to a dimensionless state by: |Rn|2×∆h/˜

nmand |[t,β·Rm]|2×∆h/˜

nm, where ∆his the mesh493

Geometric learning for shells 19

size and ˜

nmis the maximum principal component of the effective membrane stress tensor ˜

nβα at the loaded494

point of interest.495

(a) Conﬁguration at ¯

F=0.6275 (b) Conﬁguration at ¯

F=0.9425

Fig. 11: Predicted deformed shell conﬁgurations at two test snapshots with prescribed loads not included

in the training samples. Green surface indicates the reference conﬁguration (from DNS) while gray surface

indicates the conﬁguration predicted from our model.

(a) Linear momentum residual normalized,

F=0.6275

(b) Linear momentum residual normalized,

F=0.9425

normalized, ¯

F=0.6275

(d) Weighted director momentum residual

normalized, ¯

F=0.9425

Fig. 12: Normalized residual vector ﬁelds in the neighborhood of parametric coordinates (ξ,η)=(1, 0),

where concentrated force is applied.

We here want to highlight the capability of our model for capturing the nonlinearity in the response496

surface in the parametric space. To this end, we propose to compare the prediction with our model and497

the linear interpolation results, with a focus to the loading case where ¯

F=0.9425. We fetch two loading498

conditions that formulate the closest possible upper and lower bound for the targeted case ( ¯

F=0.9425),499

which could be determined as: ¯

F=0.945, 0.94. We extract deformed conﬁgurations corresponding to the500

20 Mian Xiao et al.

two fetched conditions from the database, and generate the interpolated conﬁguration by computing the501

arithmetic average of the displacement across the entire domain, as the load targeted load is the arithmetic502

average of the fetched loads. For visualization purposes, we plot the displacement prediction absolute error503

of both the interpolated conﬁguration and the predicted conﬁguration onto the initial shell conﬁguration504

in Fig. 13.

(a) abs. err. interpolated conﬁguration (b) abs. err. predicted conﬁguration

Fig. 13: Absolute error of the displacement plotted on the initial shell conﬁguration, with ¯

F=0.9425. (a)

absolute error of the interpolated conﬁguration; (a) absolute error of the predicted conﬁguration. Displace-

ment absolute error unit: m.

505

In general, we observe in Fig. 13 that the prediction of our model outperforms the result obtained by506

the linear interpolation method. We further justify the application of curvature ﬁlter with plots in Fig. 14507

and 15, as both the smoothness and the physical residual both decreases signiﬁcantly for this case.508

(a) Conﬁguration at ¯

F=0.6275 (b) Conﬁguration at ¯

F=0.9425

Fig. 14: Improvement from the curvature ﬁlter I: predicted deformed shell conﬁgurations at two test snap-

shots, where the curvature ﬁlter is not applied. Green surface indicates the groundtruth conﬁguration

while gray surface indicates the conﬁguration predicted from our model.

Geometric learning for shells 21

(a) Linear momentum residual normalized, no

curvature ﬁlter

(b) Linear momentum residual normalized,

with curvature ﬁlter

Fig. 15: Improvement from the curvature ﬁlter II: comparison between the results without curvature ﬁlter

applied and the results with curvature ﬁlter applied, with respect to the normalized residual vector ﬁelds

in the neighborhood of parametric coordinates (ξ,η) = (1, 0)at ¯

F=0.6275.

5.2 A doubly-curved shell structure509

This section provides numerical results regarding the shell conﬁguration prediction and the responses510

in a local area near the concentrated force after the resulting adjustment with physical constraints for the511

doubly-curved shell example introduced in 4.1.2. We ﬁrst present the deformed shell conﬁguration for the512

three testing paths we prescribed in 4.2, where each conﬁguration is extracted after simulated to the end513

of the loading path. As Fig. 16 shows, the predicted shell conﬁgurations mostly overlap with the reference514

shell conﬁgurations, which are from the high-ﬁdelity numerical simulation results on the ﬁne mesh shown515

in Fig. 7(c).516

(a) test path # 1 (b) test path # 2 (c) test path # 3

Fig. 16: Predicted deformed shell conﬁgurations at the last step simulation results of the three prescribed

test paths demonstrated in Section 4.2. Displacements are scaled up by 7.5 times. Green surface indicates

the groundtruth conﬁguration while gray surface indicates the conﬁguration predicted from our model.

Again, we here want to show the privilege of our model w.r.t. the linear interpolation method in higher517

dimensional parametric space. The example target loading condition and fetched loading conditions are518

summarized in Table 5. The resulting absolute error plot onto the initial conﬁguration is provided in Fig.519

17. We ﬁnd that the general magnitude and the RMSE response of the displacement absolute error of our520

model prediction are much lower than their interpolated counterparts, unlike the responses in Section 5.1.521

This further demonstrates the capability of recovering the signiﬁcant nonlinear behavior of the response522

surface in higher dimensional cases.523

Next, we would like to show the progress with solution adjustment with physical constraints. Fig. 18524

shows one example set of perturbed modes as the basis for adjusting the shell conﬁguration locally around525

the loaded points (ξ,η)=(0, −1)at the end of test path # 1. These perturbed ﬁelds are generated by526

22 Mian Xiao et al.

prediction target load condition fetched load conditions from database

cond. 1 cond. 2

(−0.94, −0.14, 0.03, 0.77) (−0.92, −0.02, −0.01, 0.75) (−0.95, −0.26, 0.08, 0.78 )

Table 5: Load conditions pfor interpolating shell conﬁguration based on the database for learning the

reduced order model.

(a) abs. err. interpolated conﬁguration (b) abs. err. predicted conﬁguration

Fig. 17: Absolute error of the displacement plotted on the initial shell conﬁguration for targeted loading in

Table 5. (a) absolute error of the interpolated conﬁguration; (a) absolute error of the predicted conﬁgura-

tion. Displacement absolute error unit: m.

random prescribe perturbation δpi(i=1, 2, 3,4) in the parametric space, while the linear independency527

is veriﬁed for the perturbed ﬁelds. Adjusted local shell conﬁgurations for three different loading paths528

are shown in Fig. 19. We also provide the computed linear momentum residual and director momentum529

residual ﬁelds within the neighborhood of the two loaded points for the three different loading cases in530

Fig. 20 ∼25.531

(a) basis # 1 and # 2 (b) basis # 3 and # 4

Fig. 18: Examples of the perturbed functional basis for optimizing the predicted shell conﬁguration locally

with physical constraints, for the case of test path # 1 around (ξ,η) = (0, −1).Green surface indicates the

non-perturbed reference local patch simulated shell conﬁguration.

Geometric learning for shells 23

(a) conﬁgurations aound (ξ,η) = (0, −1)(b) conﬁgurations aound (ξ,η) = (0, 1)

Fig. 19: Resulting local patch on the shell conﬁguration after correction based on physical constraints for

speciﬁc points of interest (e.g. concentrated load applied). Green surface indicates the reference local patch

simulated shell conﬁguration; gray surface indicates the corrected local patch.