DCEM: A deep complementary energy method for solid mechanics

TL;DR

The results demonstrate that DCEM outperforms DEM in terms of stress accuracy and efficiency and has an advantage in dealing with complex displacement boundary conditions, which is supported by theoretical analyses and numerical simulations.

Abstract

In recent years, the rapid advancement of deep learning has significantly impacted various fields, particularly in solving partial differential equations (PDEs) in the realm of solid mechanics, benefiting greatly from the remarkable approximation capabilities of neural networks. In solving PDEs, Physics-Informed Neural Networks (PINNs) and the Deep Energy Method (DEM) have garnered substantial attention. The principle of minimum potential energy and complementary energy are two important variational principles in solid mechanics. However, the well-known Deep Energy Method (DEM) is based on the principle of minimum potential energy, but there lacks the important form of minimum complementary energy. To bridge this gap, we propose the deep complementary energy method (DCEM) based on the principle of minimum complementary energy. The output function of DCEM is the stress function, which inherently satisfies the equilibrium equation. We present numerical results using the Prandtl and Airy stress functions, and compare DCEM with existing PINNs and DEM algorithms when modeling representative mechanical problems. The results demonstrate that DCEM outperforms DEM in terms of stress accuracy and efficiency and has an advantage in dealing with complex displacement boundary conditions, which is supported by theoretical analyses and numerical simulations. We extend DCEM to DCEM-Plus (DCEM-P), adding terms that satisfy partial differential equations. Furthermore, we propose a deep complementary energy operator method (DCEM-O) by combining operator learning with physical equations. Initially, we train DCEM-O using high-fidelity numerical results and then incorporate complementary energy. DCEM-P and DCEM-O further enhance the accuracy and efficiency of DCEM.

Figures (32)

• Figure 1: Schematic of DCEM: The process involves initially employing data, whether from experiments or highly accurate simulation results, to pre-train an operator learning model and obtain a good initial solution. Subsequently, the physical equations (in DCEM, we use the principle of the complementary energy, other PDEs are also feasible in the framework) are applied to refine the initial solution, enabling a faster acquisition of a reasonable reference solution.
• Figure 2: Schematic of the difference and connection between DCM and DEM, yellow circles in the Neural network are the inputs and outputs. Blue circles in the Neural network are the hidden neurons. $\text{P}()$ is the energy density of the functional. $\text{Loss}_{i}$ is the essential boundary loss. MSE is the mean square error to let the field of interest to satisfy the essential boundary. If the boundary condition is satisfied in advance, $\text{Loss}_{i}$ can be dismissed. The index i in $\text{Loss}_{i}$ means $\text{Loss}_{i}$ can be the boundary condition loss and initial condition loss if temporal problem. $\text{L}()$ is the differential operator related to the strong form of PDEs. $\lambda_{DCM}$ and $\lambda_{DEM}$ are the weight of the loss of DCM and DEM. $\lambda_{i}$ is the weight of $\text{Loss}_{i}$. Note that the number of $\lambda_{DCM}$ is determined by the number of PDEs. The dotted arrow means not all PDEs can be converted to the energy form.
• Figure 3: The relation between displacement and stress solution. The black solid line arrow is the derivation process; the dotted arrow indicates that the equation is automatically satisfied
• Figure 4: The illustration of Airy and Prandtl stress function: (a) The value of the Airy stress function: the reference point means $\phi_{A}=\partial\phi/\partial x=\partial\phi/\partial y=0$. $\bar{\boldsymbol{t}}$ is the surface force. $\boldsymbol{n}$ and $\boldsymbol{s}$ is the normal and tangential direction, respectively. $R_{x}$ ($R_{y}$) means the integral of the surface force in the x (y) direction at the boundary from the reference point to the current position B. $M$ is the moment of the current position with respect to the reference point at the boundary. Counterclockwise is the positive direction of the moment. (b) The value of the Prandtl stress function: the red lines represent the contours of the Prandtl stress function. The value of shear stress is equal to the negative gradient of the Prandtl stress function. The direction of the shear stress points is the tangent direction of the contour of the Prandtl stress function. The Prandtl stress is equal to the constant (usually taken as zero) on the boundary.
• Figure 5: The schematic of deep operator energy method based on the principle of minimum complementary energy (DCEM-O)