Performance Portable Solid Mechanics via Matrix-Free pp-Multigrid

Finite element analysis of solid mechanics is a foundational tool of modern engineering, with low-order finite element methods and assembled sparse matrices representing the industry standard for implicit analysis. We use performance models and numerical experiments to demonstrate that high-order methods greatly reduce the costs to reach engineering tolerances while enabling effective use of GPUs. We demonstrate the reliability, efficiency, and scalability of matrix-free $p$-multigrid methods with algebraic multigrid coarse solvers through large deformation hyperelastic simulations of multiscale structures. We investigate accuracy, cost, and execution time on multi-node CPU and GPU systems for moderate to large models using AMD MI250X (OLCF Crusher), NVIDIA A100 (NERSC Perlmutter), and V100 (LLNL Lassen and OLCF Summit), resulting in order of magnitude efficiency improvements over a broad range of model properties and scales. We discuss efficient matrix-free representation of Jacobians and demonstrate how automatic differentiation enables rapid development of nonlinear material models without impacting debuggability and workflows targeting GPUs

Effective Data Sampling Strategies and Boundary Condition Constraints of Physics-Informed Neural Networks for Identifying Material Properties in Solid Mechanics

Material identification is critical for understanding the relationship between mechanical properties and the associated mechanical functions. However, material identification is a challenging task, especially when the characteristic of the material is highly nonlinear in nature, as is common in biological tissue. In this work, we identify unknown material properties in continuum solid mechanics via physics-informed neural networks (PINNs). To improve the accuracy and efficiency of PINNs, we developed efficient strategies to nonuniformly sample observational data. We also investigated different approaches to enforce Dirichlet boundary conditions as soft or hard constraints. Finally, we apply the proposed methods to a diverse set of time-dependent and time-independent solid mechanic examples that span linear elastic and hyperelastic material space. The estimated material parameters achieve relative errors of less than 1%. As such, this work is relevant to diverse applications, including optimizing structural integrity and developing novel materials

Study of automatic differentiation in topology optimization

This bachelor final thesis presents a study on the integration of automatic differentiation functions into basic topology optimisation algorithms to improve not only computation speed, but also efficiency and accuracy. The main goal is to develop a fully functional automatic differentiation script capable of deriving topological expression, linear or non linear ones, aiming to find the optimal distribution. Moreover, the objectives of this research are to explore the application of automatic differentiation in fields related to topology optimisation, analyse the benefits of applying this methods and its disadvantages and review its computational efficiency. The thesis begins by introducing the fundamentals of automatic differentiation. Beforehand, during the initial stages of the project extensive practice of Matlab programming and object oriented programming was taught but it is not included in this report. A literature review is conducted to examine existing studies and approaches that utilize AD techniques. Furthermore, we dig into different AD methods, including forward mode and reverse mode, highlighting its approach with Matlab language and their advantages and limitations. Additionally, specific topologic optimisation tools and software packages commonly used are reviewed but are not included in this report. The report continues by presenting the different discretisation cases in finite element method and developing an AD based case to solve the discretisation of different 2 dimension problems. Deriving its shape functions and testing AD for a future, more sophisticated, topological problem. To achieve the main objective of performing, at least, one topology case using automatic differentiation, an AD based algorithm using different iterative methods is developed and implemented. The algorithm is tested with basic shapes and problems to be improved and more efficient, including all the possible casuistry of a mathematical expression. Ending with the research, we test different topological cases using different iterative methods. One of these methods, newton iteration method, will need an improvement to higher order gradients of the automatic differentiation algorithm. With this improvement we will test and compare both methods for several cases to conclude about the efficiency, accuracy and computing time of both iterative methods and automatic differentiation algorithm applied to topological problems. The results of the study demonstrate that automatic differentiation significantly enhances the efficiency and accuracy of topological optimisation for a certain type of problems. For these cases, AD exhibits faster convergence, improved accuracy in gradient computation, and reduced computational time. Moreover, the AD-based approach proves to be robust and applicable to not only deriving structural function, but also different problem domains, highlighting its versatility and practicality. Overall, The research highlights the potential of AD in many fields

Key issues in computational geomechanics

As stated in the introduction, the three main topics covered in this report are actual research fields. Different analyses and new developments related with these fields have been presented in the previous chapters. In the following, after a brief summary of the contributions, some directions for future research are outlined. Detailed presentations of the conclusions of each contribution are included in the corresponding sections and subsections. The most relevant contributions of this report are the following: 1. With respect to the treatment of large boundary displacements: > Quasistatic and dynamic analyses of the vane test for soft materials using a fluid–based ALE formulation and different non-newtonian constitutive laws. > The development of a solid–based ALE formulation for finite strain hyperelastic–plastic models, with applications to isochoric and non-isochoric cases. 2. Referent to the solution of nonlinear systems of equations in solid mechanics: > The use of simple and robust numerical differentiation schemes for the computation of tangent operators, including examples with several non-trivial elastoplastic constitutive laws. > The development of consistent tangent operators for substepping time–integration rules, with the application to an adaptive time–integration scheme. 3. In the field of constitutive modelling of granular materials: > The efficient numerical modelling of different problems involving elastoplastic models, including work hardening–softening models for small–strain problems and density– dependent hyperelastic–plastic models in a large–strain context. > Robust and accurate simulations of several powder compaction processes, with detailed analysis of spatial density distributions and verification of the mass conservation principle