David Tae, Kumar K. Tamma^*

CMES-Computer Modeling in Engineering & Sciences, Vol.135, No.2, pp. 843-885, 2023, DOI:10.32604/cmes.2023.023071

Abstract We propose a novel computational framework that is capable of employing different time integration algorithms and different space discretized methods such as the Finite Element Method, particle methods, and other spatial methods on a single body sub-divided into multiple subdomains. This is in conjunction with implementing the well known Generalized Single Step Single Solve (GS4) family of algorithms which encompass the entire scope of Linear Multistep algorithms that have been developed over the past 50 years or so and are second order accurate into the Differential Algebraic Equation framework. In the current state of technology, the coupling of altogether different… More >

Chein-Shan Liu ¹

CMES-Computer Modeling in Engineering & Sciences, Vol.99, No.5, pp. 371-391, 2014, DOI:10.3970/cmes.2014.099.371

Abstract We recover an unknown initial temperature for a nonlinear heat conduction equation u_t(x,t) = u_xx(x,t) + H(x,t,u,u_x), under the Cauchy boundary conditions specified on the left-boundary. The method in the present paper transforms the Cauchy problem into an inverse heat source problem to find F(x) in T_t(x,t) = T_xx(x,t) + H + F(x). By using the GL(N,R) Lie-group differential algebraic equations (LGDAE) algorithm to integrate the numerical method of lines discretized equations from sideways heat equation, we can fast recover the initial temperature and two boundary conditions on the right-boundary. The accuracy and efficiency are confirmed by comparing the exact… More >

Chein-Shan Liu¹, Satya N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.92, No.2, pp. 213-239, 2013, DOI:10.3970/cmes.2013.092.213

Abstract We show that the problem "real-time numerical differentiation" of a given noisy signal in time, by supplementing a compensated controller in the second-order robust exact differentiator, the tracking differentiator or the continuous hybrid differentiator, can be viewed as a set of differential algebraic equations (DAEs) to enhance a precise tracking of the given noisy signal. Thus, we are able to solve the highly ill-posed problem of numerical differentiation of noisy signal by using the Lie-group differential algebraic differentiators (LGDADs) based on the Lie-group GL(n,R), whose accuracy and tracking performance are better than before. The "index-two" differentiators (ITDs), which do not… More >

Chein-Shan Liu¹, Satya N. Atluri²

CMC-Computers, Materials & Continua, Vol.41, No.1, pp. 1-36, 2014, DOI:10.3970/cmc.2014.041.001

Abstract In this paper, we adopt the viewpoint of a nonlinear complementarity problem (NCP) to derive an index-one differential algebraic equations (DAEs) system for the problem of elastic-plastic wave propagation in an elastic-plastic solid undergoing small deformations. This is achieved by recasting the pointwise complementary trio in the elastic-plastic constitutive equations into an algebraic equation through the Fischer-Burmeister NCP-function. Then, for an isotropicallyhardening/ softening material under prescribed impulse loadings on a thin-walled tube with combined axial-torsional stresses, we can develop a novel algorithm based on the Lie-group differential algebraic equations (LGDAE) method to iteratively solve the resultant DAEs at each time… More >