Constitutive parameterized deep energy method for solid mechanics problems with random material parameters

Zhangyong Liang; Jihong Zhu; Huanhuan Gao

cs.LGMar 27, 2026

Constitutive parameterized deep energy method for solid mechanics problems with random material parameters

Authors: Zhangyong Liang, Jihong Zhu, Huanhuan Gao

Organizations: National Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China · Key Laboratory of CNC Equipment Reliability, Ministry of Education\National Key Laboratory of Automotive Chassis Integration and Bionics, School of Mechanical and Aerospace Engineering, Jilin University, Renmin Street 5988, 130025, Changchun, China · School of Mechanical and Aerospace Engineering, Jilin University, Renmin Street 5988, 130025, Changchun, China

Abstract

In practical structural design and solid mechanics simulations, material properties inherently exhibit random variations within bounded intervals. However, evaluating mechanical responses under continuous material uncertainty remains a persistent challenge. Traditional numerical approaches, such as the Finite Element Method (FEM), incur prohibitive computational costs as they require repeated mesh discretization and equation solving for every parametric realization. Similarly, data-driven surrogate models depend heavily on massive, high-fidelity datasets, while standard physics-informed frameworks (e.g., the Deep Energy Method) strictly demand complete retraining from scratch whenever material parameters change. To bridge this critical gap, we propose the Constitutive Parameterized Deep Energy Method (CPDEM). In this purely physics-driven framework, the strain energy density functional is reformulated by encoding a latent representation of stochastic constitutive parameters. By embedding material parameters directly into the neural network alongside spatial coordinates, CPDEM transforms conventional spatial collocation points into parameter-aware material points. Trained in an unsupervised manner via expected energy minimization over the parameter domain, the pre-trained model continuously learns the solution manifold. Consequently, it enables zero-shot, real-time inference of displacement fields for unknown material parameters without requiring any dataset generation or model retraining. The proposed method is rigorously validated across diverse benchmarks, including linear elasticity, finite-strain hyperelasticity, and complex highly nonlinear contact mechanics. To the best of our knowledge, CPDEM represents the first purely physics-driven deep learning paradigm capable of simultaneously and efficiently handling continuous multi-parameter variations in solid mechanics.

Figures & tables

Fig. 1: Material-parameter sampling and evaluation at unseen in-range parameters

(E,\nu)

for a fixed mechanical setting. Circles indicate training samples; the diamond denotes an unseen query. The depicted responses are schematic.

Fig. 2: Mechanical response family under varying material parameters. Vectors

\bm{\eta}_{1}

\bm{\eta}_{2}

, and

\bm{\eta}^{*}

specify spatially uniform properties on the same domain

\Omega

with fixed loading and boundaries

\Gamma_{D}

and

\Gamma_{N}

. Interior curves depict schematic responses.

Fig. 3: CPDEM architecture. The material encoder

g_{\theta_{p}}

maps the constitutive vector to latent features, and the coordinate encoder

g_{\theta_{c}}

maps the collocation coordinates to spatial features. Their concatenated outputs are supplied to the fusion network

g_{\theta_{g}}

, which predicts the displacement field.

Fig. 4: Material sampling and energy averaging in CPDEM. Spatial quadrature gives each material energy; their average trains one shared network. Points and field lines are schematic.

Fig. 5: Parameter coverage and error bounds for unseen parameters. (a) Distance to the nearest training parameter,

r\leq h_{\mathcal{P}_{N}}

. (b) Schematic error terms from exact-solution variation, training error, and learned-solution variation.

Problem	$\dim(\bm{\eta})$	Components of $\bm{\eta}$	$g_{\theta_{p}}$	$g_{\theta_{c}}$	$g_{\theta_{g}}$	Output dim.
2-D linear-elastic beam	2	$[E,\nu]$	$[2,16,16]$	$[2,16,16]$	$[32,32,2]$	2
2-D Neo-Hookean beam	2	$[E,\nu]$	$[2,20]$	$[2,20]$	$[40,20,2]$	2
3-D Neo-Hookean beam	2	$[E,\nu]$	$[2,30]$	$[3,30]$	$[60,30,3]$	3
3-D Neo-Hookean cuboid (torsion)	2	$[E,\nu]$	$[2,30]$	$[3,30]$	$[60,30,3]$	3
2-D Plate-with-hole $J_{2}$ plasticity	3	$[E,\nu,H]$	$[3,32,32]$	$[3,32,32]$	$[64,64,3]$	3
2-D viscoelastic creep	4	$[E_{\infty},E_{1},\xi,\nu]$	$[4,32,32]$	$[3,32,32]$	$[64,64,2]$	2

Table 1: CPDEM architecture implementation details.

Fig. 7: CPDEM and FEM displacement fields for the 2D linear elastic cantilever. Rows correspond to $(E,\nu)=(100,0.25)$ , $(223,0.30)$ , and $(500,0.35)$ . Columns show FEM displacement magnitude, CPDEM prediction, and pointwise absolute error.

Fig. 8: Training energy histories for the 2D Neo-Hookean cantilever. (a) Total potential energy; (b) internal energy; (c) external potential energy. The gray dashed line marks the Adam-to-L-BFGS transition.

Fig. 9: Training error histories for a plane-stress linear-elastic cantilever. (a) mean point error; (b) maximum point error; (c) loaded-end squared-error share. All three material parameter sets are pretraining samples.

Fig. 10: CPDEM and FEM displacement fields for the 2D Neo-Hookean cantilever. Rows correspond to $(E,\nu)=(800,0.25)$ , $(979,0.30)$ , and $(1200,0.35)$ . FEM displacement magnitude, CPDEM prediction, and pointwise absolute error.

Fig. 11: Three-dimensional Neo-Hookean bending beam. The

L=4\,\mathrm{m}

and

H=W=1\,\mathrm{m}

beam is clamped on the left and loaded by

\bar{t}_{y}=-5\,\mathrm{Pa}

on the right.

Fig. 12: CPDEM and FEM displacement fields for the 3-D Neo-Hookean cantilever. Rows correspond to $(E,\nu)=(800,0.25)$ , $(979,0.30)$ , and $(1200,0.35)$ . FEM displacement magnitude, CPDEM prediction, and pointwise absolute error.

Fig. 13: Three-dimensional hyperelastic cuboid under torsion. The left face is fixed. The right face is prescribed the $60^{\circ}$ twist in Eq. ( 46 ). Lateral tractions and body force are specified in the text.

Fig. 14: FEM and CPDEM displacement fields for the 3-D hyperelastic cuboid under torsion. Rows correspond to $(E,\nu)=(8\times 10^{5},0.25)$ , $(9.99862\times 10^{5},0.30)$ , and $(1.2\times 10^{6},0.35)$ . FEM reference, CPDEM prediction, pointwise absolute error, and deformed configuration. Each column uses a common color scale across the displayed cases.

Fig. 15: Plate with a hole: Geometry and applied displacements.

Fig. 16: Cyclic plate with a hole at peak load. Rows correspond to $(E,\nu,H)=(800,0.25,700)$ , $(1000,0.30,500)$ , and $(1200,0.35,300)$ . FEM and CPDEM displacement magnitudes, absolute error, and equivalent plastic strain.

Fig. 17: CPDEM and FEM cyclic displacement histories at interior points P1–P4:

u_{x}

(top) and

u_{y}

(bottom).

Fig. 18: CPDEM and FEM histories at P1–P4 under cyclic loading. Plastic strain (top) and von Mises stress (bottom).

Fig. 19: Viscoelastic creep of a 2-D cantilever beam. The left edge is clamped and the right edge is subjected to a horizontal, uniformly distributed tensile pressure

p=0.4\,\mathrm{Pa}

held constant for

t\geq 0

. The geometry is

L=10\,\mathrm{m}

and

h=1\,\mathrm{m}

Fig. 20: CPDEM training dynamics for viscoelastic creep at $t=10\,\mathrm{s}$ . The three rows correspond to the material parameter sets in Eq. ( 57 ). The insets show the horizontal and vertical displacement fields at the displayed snapshot.

Fig. 21: Viscoelastic creep state at $t=10\,\mathrm{s}$ : CPDEM and FEM reference. Rows correspond to the material parameter sets in Eq. ( 57 ). FEM reference, CPDEM prediction, and horizontal displacement absolute error.

Fig. 22: CPDEM and FEM creep histories at P1–P4 under constant traction. Horizontal displacement (top) and cumulative displacement increment (bottom), with zero vertical displacement at P1–P4.

Fig. 23: Stress relaxation of a 2-D cantilever beam. The left edge is clamped and the right edge is prescribed the elongation

\Delta L=0.1\,\mathrm{m}

t=0

. The geometry is

L=10\,\mathrm{m}

and

h=1\,\mathrm{m}

Fig. 24: CPDEM training energy histories for stress relaxation at $t=10\,\mathrm{s}$ . Panels (a)–(c) correspond to $(E_{\infty},E_{1},\xi,\nu)=(5,10,50,0.35)$ , $(2,10,50,0.25)$ , and $(2,10,10,0.25)$ . Insets show predicted $u_{x}$ and $u_{y}$ , and FEM $u_{y}$ .

Fig. 25: CPDEM and FEM displacement fields for stress relaxation at $t=10\,\mathrm{s}$ with $\Delta L=1\,\mathrm{m}$ . Rows correspond to $(E_{\infty},E_{1},\xi,\nu)=(5,10,50,0.35)$ , $(2,10,50,0.25)$ , and $(2,10,10,0.25)$ . FEM, CPDEM prediction, and absolute error.

Fig. 26: CPDEM and FEM relaxation histories at P1–P4: axial displacement (top) and displacement increment (bottom).

Fig. 27: Viscoelastic creep buckling of a quarter circular shell. (a) Geometry and external pressure; (b) sampling points and monitored outer surface; (c) outer-surface deformation at successive times.

Fig. 28: CPDEM training energy histories for viscoelastic creep buckling. Panels (a)–(c) correspond to $(E_{\infty},E_{1},\xi,\nu)=(4,7,20,0.35)$ , $(4,10,20,0.35)$ , and $(4,10,20,0.25)$ . Insets show $|\bm{u}|$ , $u_{r}$ , and $u_{\theta}$ .

Fig. 29: CPDEM and reference displacement fields for viscoelastic creep buckling. Rows correspond to $(E_{\infty},E_{1},\xi,\nu)=(4,7,20,0.35)$ , $(4,10,20,0.35)$ , and $(4,10,20,0.25)$ . Columns show the reference, prediction, and absolute error.

Fig. 30: CPDEM and FEM displacement histories at four monitoring points on the 2-D circle.

u_{x}

(top) and

u_{y}

(bottom).

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