We propose a direct mesh-free method for performing topology optimization by
integrating a density field approximation neural network with a displacement
field approximation neural network. We show that this direct integration
approach can give comparable results to conventional topology optimization
techniques, with an added advantage of enabling seamless integration with
post-processing software, and a potential of topology optimization with
objectives where meshing and Finite Element Analysis (FEA) may be expensive or
not suitable. Our approach (DMF-TONN) takes in as inputs the boundary
conditions and domain coordinates and finds the optimum density field for
minimizing the loss function of compliance and volume fraction constraint
violation. The mesh-free nature is enabled by a physics-informed displacement
field approximation neural network to solve the linear elasticity partial
differential equation and replace the FEA conventionally used for calculating
the compliance. We show that using a suitable Fourier Features neural network
architecture and hyperparameters, the density field approximation neural
network can learn the weights to represent the optimal density field for the
given domain and boundary conditions, by directly backpropagating the loss
gradient through the displacement field approximation neural network, and
unlike prior work there is no requirement of a sensitivity filter, optimality
criterion method, or a separate training of density network in each topology
optimization iteration.

本研究提出一种直接无网格法进行拓扑优化的方法，它能够将密度场近似神经网络与位移场近似神经网络集成起来，实现良好的优化效果，且无需使用网格法和有限元分析 (FEA)，可以无缝与后处理软件集成。

DMF-TONN：基于神经网络的直接无网格拓扑优化

DMF-TONN: Direct Mesh-free Topology Optimization using Neural Networks

We present the application of a class of deep learning, known as Physics
Informed Neural Networks (PINN), to learning and discovery in solid mechanics.
We explain how to incorporate the momentum balance and constitutive relations
into PINN, and explore in detail the application to linear elasticity, and
illustrate its extension to nonlinear problems through an example that
showcases von~Mises elastoplasticity. While common PINN algorithms are based on
training one deep neural network (DNN), we propose a multi-network model that
results in more accurate representation of the field variables. To validate the
model, we test the framework on synthetic data generated from analytical and
numerical reference solutions. We study convergence of the PINN model, and show
that Isogeometric Analysis (IGA) results in superior accuracy and convergence
characteristics compared with classic low-order Finite Element Method (FEM). We
also show the applicability of the framework for transfer learning, and find
vastly accelerated convergence during network re-training. Finally, we find
that honoring the physics leads to improved robustness: when trained only on a
few parameters, we find that the PINN model can accurately predict the solution
for a wide range of parameters new to the network---thus pointing to an
important application of this framework to sensitivity analysis and surrogate
modeling.

本文介绍了物理信息神经网络在固体力学中应用的方法，展示了通过使用多网络模型，结合动量平衡和本构关系，可以更准确地呈现一些场量变量。同时，通过测试合成数据并和解析解和数值解进行比较，验证了模型的有效性和精度，并指出了等几何分析在准确性和收敛性方面的优于有限元法的特点。我们还探索了该框架在机器学习中的应用，并发现物理信息对于提高模型的鲁棒性有很大作用。