Computer Methods in Applied Mechanics and Engineering最新文献

Solving parametric Partial Differential Equations (PDEs) for a broad range of parameters is a critical challenge in scientific computing. To this end, neural operators, which predicts the PDE solution with variable PDE parameter inputs, have been successfully used. However, the training of neural operators typically demands large training datasets, the acquisition of which can be prohibitively expensive. To address this challenge, physics-informed training can offer a cost-effective strategy. However, current physics-informed neural operators face limitations, either in handling irregular domain shapes or in in generalizing to various discrete representations of PDE parameters. In this research, we introduce a novel physics-informed model architecture which can generalize to various discrete representations of PDE parameters and irregular domain shapes. Particularly, inspired by deep operator neural networks, our model involves a discretization-independent learning of parameter embedding repeatedly, and this parameter embedding is integrated with the response embeddings through multiple compositional layers, for more expressivity. Numerical results demonstrate the accuracy and efficiency of the proposed method.

The singularly perturbed problem is characterized by the presence of narrow boundary layers, which poses challenges for traditional numerical methods due to complexity and high costs. The contemporary deep learning physics-informed neural networks (PINNs) suffer from accuracy issues while learning initial conditions, fail to capture the sharp gradient behaviors, and provide inadequate approximations to rapidly oscillating solutions. A novel technique named PATPINN is introduced to effectively address singularly perturbed parabolic problems with significant gradients in the spatio-temporal domain, utilizing a unique time and parameter multi-step asymptotic pre-training approach based on PINNs. The presented technique can assist the model in learning the system dynamic behavior and improve the accuracy of the initial conditions. It also enables PINNs to capture abrupt changes in the solution without prior knowledge of the boundary layer position, boosting its ability to approximate oscillatory solutions. This innovative approach does not require hyperparameter fine-tuning and provides a dependable deep learning approach for handling evolutionary singular perturbation problems. The proposed method is compared to PINNs and pre-training PINN (PTPINN) by solving singular convection–diffusion–reaction equations and magnetohydrodynamic equations. The results show that the proposed strategy outperforms PINNs and PTPINN in capturing the boundary layer gradient, improving the approximation accuracy and accelerating the training process, in addition to significantly improving the accuracy of PINNs in approximating the initial conditions.

This paper presents a method for performing Uncertainty Quantification in high-dimensional uncertain spaces by combining arbitrary polynomial chaos with a recently proposed scheme for sensitivity enhancement (Kantarakias and Papadakis, 2023). Including available sensitivity information offers a way to mitigate the curse of dimensionality in Polynomial Chaos Expansions (PCEs). Coupling the sensitivity enhancement to arbitrary Polynomial Chaos allows the formulation to be extended to a wide range of stochastic processes, including multi-modal, fat-tailed, and truncated probability distributions. In so doing, this work addresses two of the barriers to widespread industrial application of PCEs. The method is demonstrated for a number of synthetic test cases, including an uncertainty analysis of a Finite Element structure, determined using Topology Optimisation, with 306 uncertain inputs. We demonstrate that by exploiting sensitivity information, PCEs can feasibly be applied to such problems and through the Sobol sensitivity indices, can allow a designer to easily visualise the spatial distribution of the sensitivities within the structure.

Physics-informed neural networks (PINNs) have recently prevailed as differentiable solvers that unify forward and inverse analysis in the same formulation. However, PINNs have quite limited caliber when dealing with concentration features and discontinuous multi-material heterogeneity, hindering its application when labeled data is missing. We propose a novel physics-encoded finite element network (PEFEN) that can deal with concentration features and multi-material heterogeneity without special treatments, extra burden, or labeled data. Leveraging the interpretable discretized finite element approximation as a differentiable network in the new approach, PEFEN encodes the physics structure of multi-material heterogeneity, functional losses, and boundary conditions. We simulate three typical numerical experiments, and PEFEN is validated with a good performance of handling complex cases where conventional PINNs fail. Moreover, PEFEN entails much fewer iterations (<10%) than some published improved PINNs (namely the mixed form and domain decomposition method), and the proposed PEFEN does not employ extra variables for stresses or special treatments for subdomains. We further examine PEFEN in hyperelastic multi-layer strata with and without a pile, validating its ability for more practical applications. PEFEN is also tested for inverse analysis. In 3D experiments, transfer learning with PEFEN is validated. PEFEN need much less memory than FEM (<20%), and its training from zero initialization is faster than FEM forward analysis (>1 million dofs). It is also discussed that PEFEN may act like domain decomposition in a refined way, and a simple experiment validates that PEFEN can solve the problem with multi-scale frequency. The PEFEN, thus, proves to be a promising method and deserves further development.

Hybrid additive–subtractive manufacturing (HASM) is a revolutionary technique that, the interplay between additive and subtractive processes within an integrated machine tool allows for the fabrication of traditionally challenging complex geometries with excellent quality. However, part design for hybrid manufacturing has mostly been done by experts with rare support from computational design algorithms. Hence, the primary contribution of this work is to propose a solution for HASM-oriented structural topology optimization that incorporates both dynamic process planning and accessibility constraints. This novel optimization algorithm is developed under a unified SIMP and magic needle framework. Two sets of design variables are proposed: one for the topological description while the other for identifying the printing stage-related subdivisions. Accordingly, a series of additive manufacturing (AM) and subtractive manufacturing (SM) dedicated geometric constraints are developed based on these design variables to enable the cutting tool and laser head accessibility. Supported by the sensitivities, the structural geometry and fabrication fields can be simultaneously optimized. The effectiveness of the algorithm is proved through several numerical and experimental case studies. All the factors of cutting tool directions, HASM stages, and specific tool shapes are thorough investigated.

The Cartesian grid, which is highly popular in Computational Fluid Dynamics (CFD), has the characteristics of high mesh quality and easy generation. However, due to the limit of shape functions, the Cartesian grid with hanging nodes (CGHN) was rarely used in finite element method based CFD algorithm. Based on the framework of the immersed boundary method, a smoothed finite element method based on CGHN is developed for the fluid-structure interaction problems in incompressible fluids and large deformed structures. The gradient smoothing technique simplifies the processing of the hanging nodes and ensures the mesh density of the Cartesian elements. When solving the nonlinear N-S equations, the characteristic-based split format is combined with the stabilized pressure gradient projection to overcome the convection and pressure oscillations in the Galerkin-like method. A heterogeneous mesh mapping technology is developed for the data transfer between fluid and solid domains. An efficient, accurate and generalized mass conservation algorithm is developed to solve the pressure oscillations in data transfer between fluids and solids. The results of numerical examples show that the presented method possesses high accuracy and robustness.

The kinematics of polydisperse granular materials comprised of overlapping spheres is carefully analysed. A single-particle strain estimate is developed that summaries the deformation experienced by each particle in terms of a mean deformation gradient. This strain estimate accounts for material displaced at interparticle contacts as well as a compensatory motion of the free particle surface. Forces that are work-conjugate to the mean deformation gradient are determined; they constitute the many-body forces required for a correct mechanical behaviour in the zero-porosity limit. Notwithstanding this, pairwise interparticle forces are needed for two main reasons; they dominate the particle interactions at small overlaps and stabilise the formulation in the continuum limit. Numerical simulations are performed to demonstrate the properties of the single-particle strain estimate and to test certain aspects of the formulation. In particular, it is demonstrated that the formulation can accommodate large rotations and provides a mechanical response consistent with that of a solid material in the zero-porosity limit. It is concluded that this work forms the basis for future developments aiming at formulation of realistic contact models for plastic particles and macroscopically consistent discrete methods for granular materials.

Structural problems play a critical role in many areas of science and engineering. Their efficient and accurate solution is essential for designing and optimising civil engineering, aerospace, and materials science applications, to name a few. When appropriately tuned, Algebraic Multigrid (AMG) methods exhibit a convergence that is independent of the problem size, making them the preferred option for solving structural problems. Nevertheless, AMG faces several computational challenges, including its remarkable memory footprint, costly setup, and the relatively low arithmetic intensity of the sparse linear algebra operations. This work presents AMGR, an enhanced variant of AMG that mitigates such limitations. Its name arises from the AMG reduction framework it introduces, and its flexibility allows for leveraging several features that are common in structural problems. Namely, periodicities, spatial symmetries, and localised non-linearities. For such cases, we show how to reduce the memory footprint and setup costs of the standard AMG, as well as increase its arithmetic intensity. Despite being lighter than the standard AMG, AMGR exhibits comparable scalability and convergence rates. Numerical experiments on several industrial applications prove AMGR’s effectiveness, resulting in up to 3.7x overall speed-ups compared to the standard AMG.

Equilibrated fluid–solid-growth (FSGe) is a fast, open source, three-dimensional (3D) computational platform for simulating interactions between instantaneous hemodynamics and long-term vessel wall adaptation through mechanobiologically equilibrated growth and remodeling (G&R). Such models can capture evolving geometry, composition, and material properties in health and disease and following clinical interventions. In traditional G&R models, this feedback is modeled through highly simplified fluid solutions, neglecting local variations in blood pressure and wall shear stress (WSS). FSGe overcomes these inherent limitations by strongly coupling the 3D Navier–Stokes equations for blood flow with a 3D equilibrated constrained mixture model (CMMe) for vascular tissue G&R. CMMe allows one to predict long-term evolved mechanobiological equilibria from an original homeostatic state at a computational cost equivalent to that of a standard hyperelastic material model. In illustrative computational examples, we focus on the development of a stable aortic aneurysm in a mouse model to highlight key differences in growth patterns between FSGe and solid-only G&R models. We show that FSGe is especially important in blood vessels with asymmetric stimuli. Simulation results reveal greater local variation in fluid-derived WSS than in intramural stress (IMS). Thus, differences between FSGe and G&R models became more pronounced with the growing influence of WSS relative to pressure. Future applications in highly localized disease processes, such as for lesion formation in atherosclerosis, can now include spatial and temporal variations of WSS.

A mass-conservative high-order unfitted finite element method for convection–diffusion equations in evolving domains is proposed. The space–time method presented in [P. Hansbo, M. G. Larson, S. Zahedi, Comput. Methods Appl. Mech. Engrg. 307 (2016)] is extended to naturally achieve mass conservation by utilizing Reynolds’ transport theorem. Furthermore, by partitioning the time-dependent domain into macroelements, a more efficient stabilization procedure for the cut finite element method in time-dependent domains is presented. Numerical experiments illustrate that the method fulfills mass conservation, attains high-order convergence, and the condition number of the resulting system matrix is controlled while sparsity is increased. Problems in bulk domains as well as coupled bulk-surface problems are considered.