Computational geometry

PRELIMINARIES. Motivation: Sensitivity and Large-Scale Systems. Nonlinear Solid Mechanics: Continuous and Semi-Discretized Formulation. Concepts of Sensitivity Analysis for Linear Systems.THE SENSITIVITY OF NONLINEAR SYSTEMS. The Basic Concepts of Nonlinear Quasi-Static Problems at Regular States. Inelastic Systems. Shape Sensitivity. Buckling and Post-Buckling. Nonlinear Dynamics. Metal Forming Using the Flow Approach. Nonlinear Thermal Systems. Appendices. References. Index. Glossary of Symbols.

Parameter Sensitivity in Nonlinear Mechanics: Theory and Finite Element Computations

Foreword Chapter 1 Introduction to SHM (Daniel L Balageas) Chapter 2 Vibration-based techniques for SHM (Claus-Peter Fritzen) Chapter 3 Fiber-optics sensors (Alfredo Guemes and Jose M Menendez) Chapter 4 SHM with piezoelectric sensors (Philippe Guy and Thomas Monnier) Chapter 5 SHM using electrical resistance (Michelle Salvia and Jean-Christophe Abry) Chapter 6 Low frequency electromagnetic techniques (Michel B Lemistre) Chapter 7 Capacitive methods for SHM in civil engineering (Xavier Derobert and Jean Iaquinta) Short Bibliographies of the Contributors Index

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Structural health monitoring

(1970). Optimization Theory with Applications. Journal of the Operational Research Society: Vol. 21, No. 3, pp. 379-379.

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Optimization Theory with Applications

The transient heat conduction problems in layered materials are common encountered in practical application, but bring great challenge to most numerical algorithms. In this study, the singular boundary method (SBM), a meshless boundary collocation technique, is first employed to numerically simulate such problems based on the time-dependent fundamental solution of diffusion equation. Taking the boundary conditions and interface conditions into account, the computing system of the SBM for layered materials is established, and then is solved. The proposed method fully inherits the merits of conventional boundary-type methods while possessing its distinctive advantages. Furthermore, it is simple, straightforward, computationally efficient, and stable since it does not need to discretize temporal derivative term. Three numerical experiments are performed to verify the efficiency and accuracy of the proposed scheme. Numerical results clearly indicate the efficiency, accuracy and stability of the presented SBM for solving transient heat conduction problems in layered materials.

A meshless singular boundary method for transient heat conduction problems in layered materials

This work is devoted to some recent developments in the Higher Order Approximation introduced to the Meshless Finite Difference Method (MFDM), and its application to the solution of boundary value problems in mechanics. In the MFDM, approximation of the sought function is described in terms of nodes rather than by means of any imposed structure like elements, regular meshes etc. Therefore, the MFDM, using arbitrarily irregular clouds of nodes using the Moving Weighted Least Squares (MWLS) approximation falls into the category of the Meshless Methods (MM). The MFDM, dating to early seventies, is one of the oldest and possibly the most developed one. In this paper considered are some techniques which lead to improvement of the MFDM solution�s quality. The main objective of this paper is the presentation and overview of new ideas and the development of the Higher Order solution approach in the MFDM provided by correction terms, preceded by a brief information about the current state-of-the art of this method. The main concept of the Higher Order Approximation (HOA) used here, is based on consideration of additional terms in the local Taylor expansion of the sought function. It shall be demonstrated that such a move may essentially improve, in many ways, efficiency and solution quality of the Higher Order MFDM. The Higher Order correction terms may be applied in many aspects of the MFDM solution approach. Among them one may distinguish the a-posteriori error estimation as well as adaptive solution process with multigrid strategy. Moreover, in the present work considered are: computational implementation of the Higher Order MFDM algorithms, examination of the above mentioned aspects using 1D and 2D benchmark tests, as well as an application of the Higher Order MFDM solution approach to selected boundary value problems in mechanics.

Meshless Finite Difference Method with Higher Order Approximation—Applications in Mechanics

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.

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Selected computational aspects of the meshless finite difference method

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.

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

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.

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

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.

Sławomir Milewski

Papers

Meshless Finite Difference Method with Higher Order Approximation—Applications in Mechanics

Selected computational aspects of the meshless finite difference method

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

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

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