L. Dong¹, A. Alotaibi², S.A. Mohiuddine², S. N. Atluri³

CMES-Computer Modeling in Engineering & Sciences, Vol.99, No.1, pp. 1-85, 2014, DOI:10.3970/cmes.2014.099.001

Abstract In this expository article, a variety of computational methods, such as Collocation, Finite Volume, Finite Element, Boundary Element, MLPG (Meshless Local Petrov Galerkin), Trefftz methods, and Method of Fundamental Solutions, etc., which are often used in isolated ways in contemporary literature are presented in a unified way, and are illustrated to solve a 4^th order ordinary differential equation (beam on an elastic foundation). Both the primal formulation, which considers the 4^th order ODE with displacement as the primitive variable, as well as two types of mixed formulations (one resulting in a set of 2 second-order ODEs, and the other resulting… More >

Z. D. Han¹, S. N. Atluri^2,3

CMES-Computer Modeling in Engineering & Sciences, Vol.97, No.1, pp. 1-34, 2014, DOI:10.3970/cmes.2014.097.001

Abstract The concept of a stress tensor [for instance, the Cauchy stress σ, Cauchy (1789-1857); the first Piola-Kirchhoff stress P, Piola (1794-1850), and Kirchhoff (1824-1889); and the second Piola-Kirchhoff stress, S] plays a central role in Newtonian continuum mechanics, through a physical approach based on the conservation laws for linear and angular momenta. The pioneering work of Noether (1882-1935), and the extraordinarily seminal work of Eshelby (1916- 1981), lead to the concept of an “energy-momentum tensor” [Eshelby (1951)]. An alternate form of the “energy-momentum tensor” was also given by Eshelby (1975) by taking the two-point deformation gradient tensor as an independent… More >

TakehikoNakama¹

CMC-Computers, Materials & Continua, Vol.14, No.1, pp. 35-60, 2009, DOI:10.3970/cmc.2009.014.035

Abstract Random noise perturbs objective functions in practical optimization problems, and genetic algorithms (GAs) have been proposed as an effective optimization tool for dealing with noisy objective functions. In this paper, we investigate GAs in a variety of noisy environments where fitness perturbation can occur in any form-for example, fitness evaluations can be concurrently disturbed by additive and multiplicative noise. We reveal the convergence properties of GAs by constructing and analyzing a Markov chain that explicitly models the evolution of the algorithms in noisy environments. We compute the one-step transition probabilities of the Markov chain and show that the chain has… More >