Multidimensional Staggered Grid Residual Distribution Scheme for Lagrangian Hydrodynamics

In this work, we prove the convergence of residual distribution (RD) schemes to dissipative weak solutions of the Euler equations. We need to guarantee that the RD schemes are fulfilling the underlying structure preserving methods properties such as positivity of density and internal energy. Consequently, the RD schemes lead to a consistent and stable approximation of the Euler equations. Our result can be seen as a generalization of the Lax–Richtmyer equivalence theorem to nonlinear problems that consistency plus stability is equivalent to convergence.

This work focuses on the formulation of a four-equation model for simulating unsteady two-phase mixtures with phase transition and strong discontinuities. The main assumption consists in a homogeneous temperature, pressure and velocity fields between the two phases. Specifically, we present the extension of a residual distribution scheme to solve a four-equation two-phase system with phase transition written in a non-conservative form, i.e. in terms of internal energy instead of the classical total energy approach. This non-conservative formulation allows avoiding the classical oscillations obtained by many approaches, that might appear for the pressure profile across contact discontinuities. The proposed method relies on a finite element based residual distribution scheme which is designed for an explicit second-order time stepping. We test the non-conservative residual distribution scheme on several benchmark problems and assess the results via a cross-validation with the approximated solution obtained via a conservative approach, based on a HLLC scheme. Furthermore, we check both methods for mesh convergence and show the effective robustness on very severe test cases, that involve both problems with and without phase transition.

Residual distribution schemes for steady radiative transfer equations on unstructured meshes

Residual distribution schemes for Maxwell’s equations

SUMMARYWe are interested in the numerical approximation of steady scalar convection–diffusion problems by means of high order schemes called Residual Distribution schemes. In the inviscid case, one can develop nonlinear Residual Distribution schemes that are nonoscillatory, even in the case of very strong discontinuities, while having the most possible compact stencil, on hybrid unstructured meshes. This paper proposes and compare extensions of these schemes for the convection–diffusion problem. This methodology, in particular in terms of accuracy, is evaluated on problem with exact solutions. Its nonoscillatory behavior is tested against the Smith and Hutton problem. Copyright © 2012 John Wiley & Sons, Ltd.

Residual distribution schemes: Current status and future trends

Discretely nonlinearly stable weight-adjusted flux reconstruction high-order method for compressible flows on curvilinear grids

An ALE residual distribution scheme for the unsteady Euler equations over triangular grids with local mesh adaptation

This paper proposes a method to identify mass and stiffness matrices of structures. Element mass and stiffness matrices have mass and stiffness distributions. Therefore, a ratio between two elements in these matrices has a proper value. Element mass and stiffness matrices are divided into several groups considering this relation. A group is represented by a sparse matrix (called group matrix) with only elements in the group matrix. Mass and stiffness matrices are modified group by group. The objective function to be minimized is the sum of each norm of difference between analytical and identified group matrices. A numerical example is given to show the effectiveness of the method.

An improvement to a previously described algorithm that systematically generates shape functions is presented. The new method produces explicit closed-form shape functions which may be programmed directly. The previous method evaluated the shape functions numerically at the Gauss points by means of repetitive subroutine calls. © 1997 John Wiley & Sons, Ltd.

A new approach to optimize variable axial composite shells: A metaheuristic procedure using the finite element method

Micro-mass sensors have attracted increasing attention in the field of biomolecular and chemical detection. It has been found that the size, shape, and geometry of the structure may affect the performance of the sensor. As a result, a topology optimization methodology is proposed in this paper for the design of micro-mass sensors. A phase-field function controlled by nodal variables and finite element shape functions is used to describe the configuration of a sensor in the constructed optimization problem. The design goal is to maximize the mass detection sensitivity. On the basis of these formulations, an optimization algorithm is constructed using the finite element method and the method of moving asymptotes. Numerical examples are presented to demonstrate the validity of the proposed problem formulation. The results suggest that the performance of the micro-mass sensor can be improved by using the proposed approach.

We present the implementation of a variational finite element solver in the HelFEM program for benchmark calculations on diatomic systems. A basis set of the form is used, where (μ, ν, φ) are transformed prolate spheroidal coordinates, B n(μ) are finite element shape functions, and are spherical harmonics. The basis set allows for an arbitrary level of accuracy in calculations on diatomic molecules, which can be performed at present with either nonrelativistic Hartree‐Fock (HF) or density functional (DF) theory. Hundreds of DFs at the local spin density approximation (LDA), generalized gradient approximation (GGA), and the meta‐GGA level can be used through an interface with the Libxc library; meta‐GGA and hybrid DFs are not available in other fully numerical diatomic program packages. Finite electric fields are also supported in HelFEM, enabling access to electric properties. We introduce a powerful tool for adaptively choosing the basis set by using the core Hamiltonian as a proxy for its completeness. HelFEM and the novel basis set procedure are demonstrated by reproducing the restricted open‐shell HF limit energies of 68 diatomic molecules from the first to the fourth period with excellent agreement with literature values, despite requiring orders of magnitude fewer parameters for the wave function. Then, the electric properties of the BH and N2 molecules under finite field are studied, again yielding excellent agreement with previous HF limit values for energies, dipole moments, and dipole polarizabilities, again with much more compact wave functions than what were needed for the literature references. Finally, HF, LDA, GGA, and meta‐GGA calculations of the atomization energy of N2 are performed, demonstrating the superb accuracy of the present approach.

Estimation of mass matrix in machine tool’s weak components research by using symbolic regression

Mixed finite elements based on superconvergent patch recovery for strain gradient theory