An element-free Galerkin method which is applicable to arbitrary shapes but requires only nodal data is applied to elasticity and heat conduction problems. In this method, moving least-squares interpolants are used to construct the trial and test functions for the variational principle (weak form); the dependent variable and its gradient are continuous in the entire domain. In contrast to an earlier formulation by Nayroles and coworkers, certain key differences are introduced in the implementation to increase its accuracy. The numerical examples in this paper show that with these modifications, the method does not exhibit any volumetric locking, the rate of convergence can exceed that of finite elements significantly and a high resolution of localized steep gradients can be achieved. The moving least-squares interpolants and the choices of the weight function are also discussed in this paper.

Element‐free Galerkin methods

Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations

Built upon the two original books by Mike Crisfield and their own lecture notes, renowned scientist Rene de Borst and his team offer a thoroughly updated yet condensed edition that retains and builds upon the excellent reputation and appeal amongst students and engineers alike for which Crisfield's first edition is acclaimed. Together with numerous additions and updates, the new authors have retained the core content of the original publication, while bringing an improved focus on new developments and ideas. This edition offers the latest insights in non-linear finite element technology, including non-linear solution strategies, computational plasticity, damage mechanics, time-dependent effects, hyperelasticity and large-strain elasto-plasticity. The authors' integrated and consistent style and unrivalled engineering approach assures this book's unique position within the computational mechanics literature.

Non-Linear Finite Element Analysis of Solids and Structures: de Borst/Non-Linear Finite Element Analysis of Solids and Structures

This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

A new family of unconditionally stable one-step methods for the direct integration of the equations of structural dynamics is introduced and is shown to possess improved algorithmic damping properties which can be continuously controlled. The new methods are compared with members of the Newmark family, and the Houbolt and Wilson methods.

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Improved numerical dissipation for time integration algorithms in structural dynamics

The representation of discontinuities in analysis of blocky rock is discussed. A linkage type element is developed for addition of rock joint stiffness to the structural stiffness matrix describing the behavior of a system of rock blocks and joints. Several basic problems of jointed rock are studied. These examples demonstrate the marked influence joints may have on the stress distribution, displacements, and failure pattern of an underground opening or other structures in jointed rock. A new classification of joints is introduced, based on the application of the joint element to finite element analysis of structures in jointed rock. Normal stiffness, tangential stiffness, and shear strength are used as parameters in the classification system. The methods discussed in this paper allow a jointed rock mass to be treated as a system of blocks and links. Just as analysis of a reinforced concrete building requires detailed knowledge of the behavior of concrete alone and steel alone, the joint stiffness approach calls for and uses detailed description of the behavior of rock blocks and rock joints independently.

A Model for the Mechanics of Jointed Rock

A non-conforming element for stress analysis

A simple and efficient finite element is introduced for plate bending applications. Bilinear displacement and rotation functions are employed in conjunction with selective reduced integration. Numerical examples indicate that, despite its simplicity, the element is surprisingly accurate.

A simple and efficient finite element for plate bending

VASILESCU SUGGESTS THAT ROCK MECHANICS IS A NATURAL SCIENCE, ALTHOUGH THE AUTHOR IN THE CONTEXT OF THE PROBLEM BEING DISCUSSED, CONSIDERS THAT ROCK MECHANICS IS A BRANCH OF ENGINEERING, AND MATHEMATICALLY BASED METHODS OF DESCRIBING ROCK BEHAVIOR ARE NOT ONLY APPROPRIATE, BUT EXTREMELY POWERFUL. IN THE DESIGN OF EXCAVATIONS IN JOINT ROCK, ROCK MECHANICS IS VERY USEFUL. TERZAGHI OFTEN EXPRESSED THE VIEW THAT GEOLOGICAL UNCERTAINTIES LIMITED THE RELIABILITY OF ANALYSES IN WHICH A PARTICULAR JOINT WAS ASSUMED TO OCCUR AT A GIVEN LOCATION. THE MATHEMATICAL MODEL ENABLES THE EXTREME CASES TO BE STUDIED THROUGH PARAMETER STUDIES AND STATISTICAL METHODS. REPORTS ISSUED: A MODEL FOR THE MECHANICS OF JOINTED ROCK, RICHARD E. GOODMAN, ROBERT L. TAYLOR, TOR L. BREKKE, ASCE PROC. PAPER 5937, MAY 1968.

Robert L. Taylor

Papers

Improved numerical dissipation for time integration algorithms in structural dynamics

A Model for the Mechanics of Jointed Rock

A non-conforming element for stress analysis

A simple and efficient finite element for plate bending

Closure on a model for the mechanics of jointed rock