Static Finite Element Analysis of Hyperelastic Materials via Hamiltonian Simulation Using Carleman Linearization
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Nonlinear finite element analysis of hyperelastic materials is essential for simulating large deformations. However, conventional methods like the Newton-Raphson method need to build a stiffness matrix and calculate its inverse at every calculation step. This leads to high computational costs and can be a bottleneck, especially for large-scale problems. To address this problem, we propose a new quantum computing-based method. In this method, we transform the static equilibrium equation into a nonlinear time evolution equation. We then convert this into a system of linear equations using Carleman linearization by embedding the nonlinear terms into a higher-dimensional tensor space. Subsequently, we find the equilibrium state using Hamiltonian simulation. Here, the hyperelastic constitutive law is expressed as a polynomial using a surrogate model based on Radial Basis Function (RBF) interpolation. The proposed method was applied to a Neo-Hookean hyperelastic material model, and its validity was assessed by comparing the results with those obtained using existing iterative solvers. We also investigated how the solution is affected by the polynomial approximation of the constitutive law and the truncation order of Carleman linearization. The results showed that by increasing the approximation accuracy, we could obtain results close to those of conventional classical methods.
