Detecting hidden structures from a static loading experiment: topology optimization meets physics-informed neural networks

TL;DR

This work introduces a topology optimization framework based on PINNs that identifies concealed geometries using exposed surface data from a single loading experiment, without prior knowledge of the number or types of shapes.

Abstract

Most noninvasive imaging techniques utilize electromagnetic or acoustic waves originating from multiple locations and directions to identify hidden geometrical structures. Surprisingly, it is also possible to image hidden voids and inclusions buried within an object using a single static thermal or mechanical loading experiment by observing the response of the exposed surface of the body, but this problem is challenging to invert. Although physics-informed neural networks (PINNs) have shown promise as a simple-yet-powerful tool for problem inversion, they have not yet been applied to imaging problems with a priori unknown topology. Here, we introduce a topology optimization framework based on PINNs that identifies concealed geometries using exposed surface data from a single loading experiment, without prior knowledge of the number or types of shapes. We allow for arbitrary solution topology by representing the geometry using a material density field combined with a novel eikonal regularization technique. We validate our framework by detecting the number, locations, and shapes of hidden voids and inclusions in many example cases, in both 2D and 3D, and we demonstrate the method's robustness to noise and sparsity in the data. Our methodology opens a pathway for PINNs to solve geometry optimization problems in engineering.

Figures (7)

• Figure 1: Setup of two prototypical geometry identification problems in elastic bodies under mechanical loading.a, A square elastic matrix with hidden voids or inclusions is pulled by a known uniform traction on two opposite sides. The goal is to identify the number, locations, and shapes of the voids or inclusions within using measurements of the displacement occurring along the outer boundary of the matrix. b, An elastic layer on top of a hidden rigid substrate is compressed from the top by a uniform pressure. The goal is to identify the shape of the substrate using measurements of the displacement of the top surface.
• Figure 2: TO framework for noninvasive detection of hidden geometries. The geometry of the system, which is initially unknown, is parameterized by a material density field given through a level-set function and equal to 1 in the elastic body and 0 in the voids or inclusions. The level-set function and the physical quantities describing the problem are approximated with deep neural networks designed to inherently satisfy the applied BCs. These neural networks are then trained to minimize a loss function that drives the material density and physical quantities towards satisfying the governing equations of the problem while matching discrete surface measurements. A crucial eikonal regularization term in the loss function ensures that the material density transitions between 0 and 1 over a prescribed length scale and avoids settling on intermediate values. By the end of the optimization, the converged material distribution reveals the location and shapes of the hidden structures.
• Figure 3: Eikonal regularization of the material density.$\mathbf{a}$, A random level-set function $\phi$ yields a material density $\rho = \mathrm{sigmoid}(\phi/\delta)$ with large regions of values between 0 and 1, due to the nonuniformity of the gradient $|\nabla \phi|$ along the material boundaries defined by the zero level-set of $\phi$ (black lines). $\mathbf{b}$, Constraining $\phi$ to solve the eikonal equation $\nabla \phi = 1$ in a narrow band $\Omega_\mathrm{eik}$ of thickness $w$ (delineated by the dashed lines) along the material boundaries results in a uniform transition thickness of $\rho$ from 0 to 1, without large regions of intermediate density values. $\mathbf{c}$, The loss $\mathcal{L}_\mathrm{eik}$ implements the eikonal regularization in the PINN-based TO framework by penalizing violations of the constraint $|\nabla \phi| = 1$ in a subset of collocation points $\Omega_\mathrm{eik}^d \subset \Omega_d$ that approximates the true narrow band $\Omega_\mathrm{eik}$.
• Figure 4: Identification of voids and inclusions in elastic matrices. A linear elastic matrix containing voids (a-d): a, The various loss components that enforce the solution to match the surface measurement data, satisfy the governing equations, and obey the eikonal regularization, are being minimized during the training process. b,c, The final level-set function $\phi$ and its gradient magnitude $|\nabla \phi|$ show the effect of the eikonal regularization, making $\phi$ a signed distance function in narrow band along the interface. d, The final material density $\rho$ reveals the number, locations, and shapes of the hidden voids, which are compared with the ground truth shown in dotted white lines. e, The final material density predictions in the case of a linear elastic matrix containing soft, stiff or rigid inclusions. f, The final material density predictions in the case of a nonlinear hyperelastic matrix containing voids subject to large stretches.
• Figure 5: Identification of substrate shape underneath a periodic linear elastic layer.a, The various loss components that enforce the solution to match the surface measurement data, satisfy the governing equations, and obey the eikonal regularization, are being minimized during the training process. b,c, The final level-set function $\phi$ and its gradient magnitude $|\nabla \phi|$ show the effect of the eikonal regularization, which makes $\phi$ a signed distance function in narrow band along the material boundary. d, The final material density $\rho$ reveals the shape of the buried rigid substrate.