Application of the RBF collocation method to transient coupled thermoelasticity

Abstract

Purpose In this study, the authors aim to upgrade their previous developments of the local radial basis function collocation method (LRBFCM) for heat transfer, fluid flow, electromagnetic problems and linear thermoelasticity to dynamic-coupled thermoelasticity problems. Design/methodology/approach The authors solve a thermoelastic benchmark by considering a linear thermoelastic plate under thermal and pressure shock. Spatial discretization is performed by a local collocation with multi-quadrics augmented by monomials. The implicit Euler formula is used to perform the time stepping. The system of equations obtained from the formula is solved using a Newton–Raphson algorithm with GMRES to iteratively obtain the solution. The LRBFCM solution is compared with the reference finite-element method (FEM) solution and, in one case, with a solution obtained using the meshless local Petrov–Galerkin method. Findings The performance of the LRBFCM is found to be comparable to the FEM, with some differences near the tip of the shock front. The LRBFCM appears to converge to the mesh-converged solution more smoothly than the FEM. Also, the LRBFCM seems to perform better than the MLPG in the studied case. Research limitations/implications The performance of the LRBFCM near the tip of the shock front appears to be suboptimal because it does not capture the shock front as well as the FEM. With the exception of a solution obtained using the meshless local Petrov–Galerkin method, there is no other high-quality reference solution for the considered problem in the literature yet. In most cases, therefore, the authors are able to compare only two mesh-converged solutions obtained by the authors using two different discretization methods. The shock-capturing capabilities of the method should be studied in more detail. Originality/value For the first time, the LRBFCM has been applied to problems of coupled thermoelasticity.

Accurate Stabilization Techniques for Rbf-Fd Meshless Discretizations with Neumann Boundary Conditions

“…Equation (18), which is the discrete version of the PDE Lu = f we started with, can now be solved for u h ∈ R N I . We point out that, if M is ill conditioned in equation (7), so is M T of equation (17). We will show that the onset of instabilities at the boundary is due to ill conditioning of the interpolation matrix M, and that such ill conditioning is essentially influenced by the direction of the boundary normals appearing in the corresponding rows, see equations (8) and (9).…”

“…Neumann BCs can be enforced both at the stencil level, i.e., when the local interpolants exactly satisfy the BCs at the boundary nodes [16,17,18,10,19], or at the assembly level, i.e., when the equations for the BCs at the boundary nodes appear explicitly in the final sparse matrix [20,21]. In both cases the RBF-FD collocation method, however, may lead to large errors and/or ill-conditioning problems.…”

A spacetime collocation Trefftz method for solving the inverse heat conduction problem

“…A is a matrix of the Trefftz basis functions with the size of I 3 O, k with the size of O 3 1 is a vector of unknown coefficients, B with the size of I 3 1 is a vector of given values from Dirichlet boundary condition at collocation points; and I is the number of collocation points, O is the number of the terms related to the order of the basis function as depicted in equation (15), which can be defined as…”

“…Some meshless methods have evoked considerable interest for finding solutions of initial value problems and boundary value problems such as heat conduction problems. A significant number of meshless methods have been proposed, such as the Trefftz method, 9–12 the method of fundamental solution (MFS), 13,14 the radial basis functions method, 15,16 the modified polynomial expansion algorithm, 17 and the Local Petrov–Galerkin Method. 18 The collocation Trefftz method is one of the boundary-type meshless methods for solving boundary value problems where approximate solutions are expressed as a linear combination of basis functions.…”

A meshfree method for analysis of thermo-elastic problems with moving heat sources in welding

“…Strong-form mesh-free methods that have been used successfully for these kinds of problems include the Local Radial Basis Function Collocation Method (LRBFCM) which has been applied to analyze thermomechanical problems in transient coupled thermoelasticity and Hot Rolling [14][15], as well as the Smoothed Particle Hydrodynamics (SPH) which has been used for the modelling of metal forging and thermomechanical processes in break systems [16][17].…”