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• Node's residual descent method (NRDM) for linear elasticity boundary value problem. • Solved the second-order PDE boundary value problem with the first-order GFD scheme. • The NRDM eliminates the task of assembling large global sparse matrices. • The computational accuracy and convergence rate are greatly improved in the NRDM. • The NRDM retain all the advantages of GFDM. This study proposes a node's residual descent method (NRDM) for the linear elasticity boundary value problem. In this method, the boundary information and deformation states are attached to the nodes, and the partial differential equations (PDEs) are then transformed into a generalized difference form with the first-order generalized finite difference (GFD) scheme. Furthermore, this study thoroughly describes the NRDM, confirms the accuracy and convergence by numerical examples, performs an error analysis, and discusses the influence of computational parameters. Results demonstrate that NRDM has high computational accuracy and superliner convergence rate. The double derivation is implemented to solve the second-order PDE boundary value problem with the first-order GFD scheme, thus simplifying the requirement for star connectivity and reducing the computational complexity. NRDM is a matrix-free iterative method that eliminates the task of assembling large global sparse matrices, thus avoiding the numerous matrix–matrix product operations. Rather than solving large systems of linear equations, many extremely small matrix multiplications are present during iterations, which are extremely friendly to parallel computing and can be adapted to the computation of ultra-multi-degree-of-freedom problems. The computational accuracy and convergence rate are considerably improved in NRDM compared to the classical GFD method. [ABSTRACT FROM AUTHOR]

• Numerical method for electrostatic and electroelastic problems. • Explicitly handling the electroelastic coupling issues. • Local basis coordinate technique for any polarization directions. • Achieving second-order accuracy for multiphysics field variables. Piezoelectric materials are extensively used in engineering for the fabrication of sensors, transducers, and actuators. Due to the coupling characteristics, anisotropy, and arbitrariness of polarization directions, the tasks of mesh generation, numerical integration, and global equation formulation involved in numerical computations are complex and nontrivial. To solve electrostatic and electroelastic problems, an easily implementable node's residual descent method (NRDM) is established in this study. The capability and accuracy of NRDM are validated through the solution of three-dimensional electrostatic, linear elastic, and electroelastic problems, achieving relative errors of 0.17%–1.60% for stress and 0.02%–0.15% for other variables. Fundamental and higher-order variables exhibit second-order accuracy, with convergence rates ranging from 1.98 to 2.27 using a first-order generalized finite difference scheme. This comprehensively validates the double derivation technique. The electroelastic coupling problems can be addressed by explicitly calculating electric displacement and stress based on the stress–charge form of piezoelectric constitutive equations during the iteration process. Additionally, the local basis coordinate technique is seamlessly integrated, enabling the solution of anisotropic piezoelectric problems with arbitrary polarization directions through explicit tensor operations. [ABSTRACT FROM AUTHOR]

Abstract: This work is dedicated to the presentation of a matrix-free method for global linear stability analysis in geometries composed of multi-connected rectangular subdomains. An Arnoldi technique using snapshots in subdomains of the entire geometry combined with a multidomain linearized Direct Numerical Finite difference simulations based on an influence matrix for partitioning are adopted. The method is illustrated by three benchmark problems: the lid-driven cavity, the square cylinder and the open cavity flow. The efficiency of the method to extract large-scale structures in a multidomain framework is emphasized. The possibility to use subset of the full domain to recover the perturbation associated with the entire flow field is also highlighted. Such a method appears thus a promising tool to deal with large computational domains and three-dimensionality within a parallel architecture. [Copyright &y& Elsevier]

Abstract: We present a memory-efficient and parallel framework for finite element operator application implemented in the generic open-source library deal.II. Instead of assembling a sparse matrix and using it for matrix–vector products, the operation is applied by cell-wise quadrature. The evaluation of shape functions is implemented with a sum-factorization approach. Our implementation is parallelized on three levels to exploit modern supercomputer architecture in an optimal way: MPI over remote nodes, thread parallelization with dynamic task scheduling within the nodes, and explicit vectorization for utilizing processors’ vector units. Special data structures are designed for high performance and to keep the memory requirements to a minimum. The framework handles adaptively refined meshes and systems of partial differential equations. We provide performance tests for both linear and nonlinear PDEs which show that our cell-based implementation is faster than sparse matrix–vector products for polynomial order two and higher on hexahedral elements and yields ten times higher Gflops rates. [Copyright &y& Elsevier]

In this study, a matrix-free implicit dual time-stepping method has been developed. It is implemented, together with a low-Reynolds-number q–ω turbulence model, in a high-order upwind finite-volume solver on unstructured grids. Semi-implicit treatment of the source terms of the q and ω equations is also introduced to further stabilize the numerical solution. It has been found that these techniques provide strong stabilization in the computation of a supersonic flow with complex shock–boundary-layer interactions in a channel with a backward-facing step. The proposed method has a low-memory overhead, similar to an explicit scheme, while it shows good stability and computational efficiency as an implicit scheme. The method developed has been validated by comparing the computed results with the corresponding experimental measurements and other calculated results, which shows good agreement. Research is being done to extend the method to calculate unsteady turbulent flows. [Copyright &y& Elsevier]