Approximation d'un problème biharmonique par élément fini P1

International audienceWe propose an approximation of the solution of the biharmonic problem in $H_0^2(\Omega)$ which relies on the discretization of the Laplace operator using nonconforming continuous piecewise linear finite elements.Nous proposons une approximation de la solution du problème bi-harmonique dans $H_0^2(\Omega)$ basée sur la discrétisation du Laplacien par éléments finis P1 continus mais non conformes

Localized Method of Approximate Particular Solutions for Solving Fourth-Order PDEs

In the past, dealing with fourth-order partial differential equations using the Local Method was not reliable due to difficulties in solving them directly. An approach such as splitting these equations into two Poisson differential equations was adopted to alleviate such challenges. However, this has a limitation since it is only applicable to Dirichlet and Laplace boundary conditions. In this paper, we solve fourth-order PDEs directly using the LMAPS. The improvement on the accuracy of this method was as a result of the proposed distribution of boundary conditions to alternating boundary points. And, also the use of suitable shape parameter; calculated using LOOCV(Leave-One-Out-Cross-Validation) Algorithm. The effectiveness of this Method was evident when we compared the results from two numerical examples

The development of a program of calculation to determine the heat distribution in multilayered plates

Разработана методика и программа расчета для численного моделирования распределения тепла в многослойных тонких пластинах на основе решения одномерного не стационарного уравнения теплопроводности. Предполагается, что межд

Radial Basis Function Differential Quadrature Method for the Numerical Solution of Partial Differential Equations

In the numerical solution of partial differential equations (PDEs), there is a need for solving large scale problems. The Radial Basis Function Differential Quadrature (RBFDQ) method and local RBF-DQ method are applied for the solutions of boundary value problems in annular domains governed by the Poisson equation, inhomogeneous biharmonic equation, and the inhomogeneous Cauchy-Navier equations of elasticity. By choosing the collocation points properly, linear systems can be obtained so that the coefficient matrices have block circulant structures. The resulting systems can be efficiently solved using matrix decomposition algorithms (MDAs) and fast Fourier transforms (FFTs). For the local RBFDQ method, the MDAs used are modified to account for the sparsity of the arrays involved in the discretization. An adjusted Fasshauer estimate is used to obtain a good shape parameter value in the applied radial basis functions (RBFs) for the global RBF-DQ method while the leave-one-out cross validation (LOOCV) algorithm is employed for the local RBF-DQ method using a sample of local influence domains. A modification of the kdtree algorithm is used to select the nearest centers for each local domain. In several numerical experiments, it is shown that the proposed algorithms are capable of solving large scale problems while maintaining high accuracy

Adaptive isogeometric methods with C1C^1 (truncated) hierarchical splines on planar multi-patch domains

Isogeometric analysis is a powerful paradigm which exploits the high smoothness of splines for the numerical solution of high order partial differential equations. However, the tensor-product structure of standard multivariate B-spline models is not well suited for the representation of complex geometries, and to maintain high continuity on general domains special constructions on multi-patch geometries must be used. In this paper we focus on adaptive isogeometric methods with hierarchical splines, and extend the construction of $C^1$ isogeometric spline spaces on multi-patch planar domains to the hierarchical setting. We introduce a new abstract framework for the definition of hierarchical splines, which replaces the hypothesis of local linear independence for the basis of each level by a weaker assumption. We also develop a refinement algorithm that guarantees that the assumption is fulfilled by $C^1$ splines on certain suitably graded hierarchical multi-patch mesh configurations, and prove that it has linear complexity. The performance of the adaptive method is tested by solving the Poisson and the biharmonic problems

Parallel computations based on domain decompositions and integrated radial basis functions for fluid flow problems

The thesis reports a contribution to the development of parallel algorithms based on Domain Decomposition (DD) method and Compact Local Integrated Radial Basis Function (CLIRBF) method. This development aims to solve large scale fluid flow problems more efficiently by using parallel high performance computing (HPC). With the help of the DD method, one big problem can be separated into sub-problems and solved on parallel machines. In terms of numerical analysis, for each sub-problem, the overall condition number of the system matrix is significantly reduced. This is one of the main reasons for the stability, high accuracy and efficiency of parallel algorithms. The developed methods have been successfully applied to solve several benchmark problems with both rectangular and non-rectangular boundaries. In parallel computation, there is a challenge called Distributed Termination Detection (DTD) problem. DTD concerns the discovery whether all processes in a distributed system have finished their job. In a distributed system, this problem is not a trivial problem because there is neither a global synchronised clock nor a shared memory. Taking into account the specific requirement of parallel algorithms, a new algorithm is proposed and called the Bitmap DTD. This algorithm is designed to work with DD method for solving Partial Differential Equations (PDEs). The Bitmap DTD algorithm is inspired by the Credit/Recovery DTD class (or weight-throw). The distinguishing feature of this algorithm is the use of a bitmap to carry the snapshot of the system from process to process. The proposed algorithm possesses characteristics as follows. (i) It allows any process to detect termination (symmetry); (ii) it does not require any central control agent (decentralisation); (iii) termination detection delay is of the order of the diameter of the network; and (iv) the message complexity of the proposed algorithm is optimal. In the first attempt, the combination of the DD method and CLIRBF based collocation approach yields an effective parallel algorithm to solve PDEs. This approach has enabled not only the problem to be solved separately in each subdomain by a Central Processing Unit (CPU) but also compact local stencils to be independently treated. The present algorithm has achieved high throughput in solving large scale problems. The procedure is illustrated by several numerical examples including the benchmark lid-driven cavity flow problem. A new parallel algorithm is developed using the Control Volume Method (CVM) for the solution of PDEs. The goal is to develop an efficient parallel algorithm especially for problems with non-rectangular domains. When combined with CLIRBF approach, the resultant method can produce high-order accuracy and economical solution for problems with complex boundary. The algorithm is verified by solving two benchmark problems including the square lid-driven cavity flow and the triangular lid-driven cavity flow. In both cases, the accuracy is in great agreement with benchmark values. In terms of efficiency, the results show that the method has a very high efficiency profile and for some specific cases a super-linear speed-up is achieved. Although overlapping method yields a straightforward implementation and stable convergence, overlapping of sub-domains makes it less applicable for complex domains. The method even generates more computing overhead for each subdomain as the overlapping area grows. Hence, a parallel algorithm based on non-overlapping DD and CLIRBF has been developed for solving Navier-Stokes equations where a CLIRBF scheme is used to solve the problem in each subdomain. A relaxation factor is employed for the transmission conditions at the interface of sub-domains to ensure the convergence of the iterative method while the Bitmap DTD algorithm is used to achieve the global termination. The parallel algorithm is demonstrated through two fluid flow problems, namely the natural convection in concentric annuli (Boussinesq fluids) and the lid-driven cavity flow (viscous fluids). The results confirm the high efficiency of the present method in comparison with a sequential algorithm. A super-linear efficiency is also observed for a range of numbers of CPUs. Finally, when comparing the overlapping and non-overlapping parallel algorithms, it is found that the non-overlapping one is less stable. The numerical results show that the non-overlapping method is not able to converge for high Reynolds number while overlapping method reaches the same convergence profile as the sequential CLIRBF method. Thus, in this research when dealing with non-Newtonian fluids and large scale problems, the overlapping method is preferred to the nonoverlapping one. The flow of Oldroyd-B fluid through a planar contraction was considered as a benchmark problem. In this problem, the singularity of stress at the re-entrant corners always poses difficulty to numerical methods in obtaining stable solutions at high Weissenberg numbers. In this work, a high resolution simulation of the flow is obtained and the contour of streamline is shown to be in great agreement with other results