Sławomir Milewski

The effective interface approach for coupling of the FE and meshless FD methods and applying essential boundary conditions

The paper presents a new effective technique for coupling two computational methods with different types of discretization and approximation. It is based on a concept of two adjacent subdomains which are connected with each other by means of a thin layer of material. Each of the subdomain may have a different discretization structure and approximation base. The standard Finite Element Method (FEM) as well as the meshless Finite Difference Method (MFDM) are applied here to be coupled. However, the coupling can be used for any other methods. The same concept is used for applying the essential boundary conditions for both of the methods. The width of the interface layer, depending on discretization density, is evaluated by means of several heuristic assumptions. The paper is illustrated with selected two- and three-dimensional examples.

Selected computational aspects of the meshless finite difference method

Meshless Finite Difference Method (MFDM) is nowadays a powerful engineering tool for numerical analysis of boundary value problems. Nowadays, its computational capabilities are not fully used mainly due to the lack of suitable commercial software. This paper briefly presents current state-of-the-art of the MFDM solution approach as well as deals with the selected computational aspects of the MFDM. A set of Matlab functions written by the author is attached to the paper. Techniques for generation of nodes, MFD stars, formulas, equations as well as local approximation technique and numerical integration schemes are discussed there.

In search of optimal acceleration approach to iterative solution methods of simultaneous algebraic equations

This paper presents several new proposals for acceleration of iterative solution methods of both linear and non-linear Simultaneous Algebraic Equations (SAE). The main concept is based on the successive over-relaxation technique (SOR). A new simple and effective way of evaluation of the relaxation parameter is based on either minimization or annihilation of the subsequent solution residuum. The other concept effectively uses features of the infinite geometrical progression. Its quotient is built using solution increments in several initial series of subsequent iterative steps. Both acceleration mechanisms were also combined in order to obtain the best acceleration of the solution process for Simultaneous Linear Algebraic Equations (SLAE). These concepts were tested on many 1D and 2D benchmark problems, with banded and/or sparse systems. For the relaxed Gauss-Seidel (G-S) approach, the convergence rates were up to 200 times better when compared with the standard G-S one. Significant convergence improvement was also reached while testing non-linear SAEs (with the relaxed Newton-Raphson method). The numerical models of the selected engineering problems were based on the meshless approach, due to their more sophisticated nature (when compared e.g. with the finite element analysis).

Coupling finite element method with meshless finite difference method in thermomechanical problems

This paper focuses on coupling two different computational approaches, namely finite element method (FEM) and meshless finite difference method (MFDM), in one domain. The coupled approach is applied in solving thermomechanical initialboundary value problem where the heat transport in the domain is non-stationary. In this method, the domain is divided into two subdomains for FEM and MFDM, respectively. Contrary to other coupling techniques, the approach presented in this paper is defined in terms of mathematical problem formulation rather than at the approximation level. In the weak form of thermomechanical initialboundary value problem (variational principle), the appropriate additional coupling integrals are defined a-priori. Subsequently, the FEM and the MFDM approximations, which may differ from each other, are provided to the formulation. It is assumed that there exists a very thin layer of material between the subdomains, which is not spatially discretized. The width of this layer may be considered the coupling parameter and it is the same for both, thermal and mechanical parts. Similar approach is applied to essential boundary conditions (e.g. prescribed temperature and displacements). Consequently, the consistent formulation of the mixed problem for the coupled FEMMFDM method is derived. The analysis is illustrated with two- and three-dimensional examples of mechanical and thermomechanical problems.

A’posteriori Error Estimation Based on Higher Order Approximation in the Meshless Finite Difference Method

The paper presents recent developments in the Higher Order Approximation applied to the Meshless Finite Difference Method MFDM [13]. The concept of the Higher Order Approximation (HOA) [14] is based on considering additional terms in the Taylor expansion of the searched function. Those terms may consist of HO derivatives as well as their jumps and/or singularities. They are used as correction terms to the standard meshless FD operator. Among many applications of the HOA, special emphasis is focused on a’posteriori estimation of the solution and the residual error in both the local and global forms. Thus the HOA approach provides results which may be also used as a high quality reference solution in global or local error estimators. A variety of 1D and 2D tests done indicate clear superiority of such estimation approach over those currently used in the other discrete methods [1].

Global-local Petrov-Galerkin formulations in the Meshless Finite Difference Method

The paper presents the recent developments in both the Local Petrov- Galerkin (LPG) formulations of the boundary value problems of mechanics, and the Meshless Finite Difference Method MFDM of numerical analysis. The MLPG formulations use the well-known concept of the Petrov-Galerkin weak approach, where the test function may be different from the trial function. The support of such test function is limited to chosen subdomains, usually of regular shape, rather than to the whole domain. This significantly simplifies the numerical integration. MLPG discretization is performed here for the first time ever, in combination with the MFDM, the oldest and possibly the most developed meshless method. It is based on arbitrarily irregular clouds of nodes and moving weighted least squares approximation (MWLS), using here additional Higher Order correction terms. These Higher Order terms, originated from the Taylor series expansion, are considered in order to raise the local approximation rank in the most efficient manner, as well as to estimate both the a-posteriori solution and residual errors. Some new concepts of development of the original MLPG formulations are proposed as well. Several benchmark problems are analysed. Results of preliminary tests are very encouraging.

Determination of the truss static state by means of the combined FE/GA approach, on the basis of strain and displacement measurements

ABSTRACT A real engineering inverse problem is considered in this paper, namely the recovery of the full static state of plane truss for Structural Health Monitoring (SHM) purposes. Available experimental data include strain or strain/displacement measurements done at few selected construction bars and nodes. All measurements are performed by means of sensors attached to the construction. The appropriate non-linear constrained optimization problem is formulated. Its numerical analysis is performed through the combination of the finite element method with genetic algorithms (FE/GA). Apart from the truss static load determination, the proposed approach may be applied to the identification of its bars with reduced tensile stiffness, as well as defective strain sensors. Moreover, the selection of the optimal number of strain sensors is considered. The work is illustrated with selected examples of truss analysis with various geometric complexity.

Combination of the meshless finite difference approach with the Monte Carlo random walk technique for solution of elliptic problems

Abstract This paper proposes a stochastic approach for the fast and effective numerical analysis of the second order elliptic differential equations. It is based upon the well-known Monte Carlo (MC) method with a random walk (RW) technique, carried out on the grid of points. This method allows for accurate estimation of the solution of the differential equation at selected point(s) of the domain and/or its boundary. It extends the standard formulation of the Monte Carlo–random walk (MC–RW) approach by means of its appropriate combination with the meshless version of the finite difference method. In this manner, the proposed approach may deal with elliptic equations in more general non-homogeneous form as well as boundary conditions of both essential and natural types. Moreover, arbitrarily irregular clouds of nodes may be used, with no a-priori imposed nodes structure. Therefore, the meshless MC/RW approach may be applied to the significantly wider class of problems with more complex geometry. This concept was examined on variety of 2D boundary value problems. Selected numerical results are presented and discussed. A simple Matlab code is included as well.

Higher order meshless schemes applied to the finite element method in elliptic problems

Abstract This paper presents selected approximation techniques, typical for the meshless finite difference method (MFDM), although applied to the finite element method (FEM). Finite elements with standard or hierarchical shape functions are coupled with higher order meshless schemes, based upon the correction terms of a simple difference operator. Those terms consist of higher order derivatives, which are evaluated by means of the appropriate formulas composition as well as a numerical solution, which corresponds to the primary interpolation order, assigned to element shape functions. Correction terms modify the right-hand sides of algebraic FE equations only, yielding an iterative procedure. Therefore, neither re-generation of the stiffness matrix nor introduction of any additional nodes and/or degrees of freedom is required. Such improved FE-MFD solution approach allows for the optimal application of advantages of both methods, for instance, a high accuracy of the nodal FE solution and a derivatives’ super-convergence phenomenon at arbitrary domain points, typical for the meshless FDM. Existing and proposed higher order techniques, applied in the FEM, are compared with each other in terms of the solution accuracy, algorithm efficiency and computational complexity. In order to examine the considered algorithms, numerical results of several two-dimensional benchmark elliptic problems are presented. Both the accuracy of a solution and the solution’s derivatives as well as their convergence rates, evaluated on irregular and structured meshes as well as arbitrarily irregular adaptive clouds of nodes, are taken into account.

Numerical modelling of slumps under highways located on a mining damage area, based on experimental measurements

Abstract A system of ground movement measurements under a highway in an Upper Silesia (Poland, Europe) mining damage area has been presented. The objective of the research was to develop a numerical analysis of such slumps occurring under a highway, carried out by a reliable and fast method of converting the results of relevant experimental measurements into ground displacements under the road. For engineering purposes, the results of both monitoring and analysis should be proceeded online, in a fast, sufficiently accurate and reliable way.