Delfim Soares Jr.¹

CMES-Computer Modeling in Engineering & Sciences, Vol.105, No.5, pp. 341-360, 2015, DOI:10.3970/cmes.2015.105.341

Abstract In this work, a second-order time-marching procedure for dynamics is discussed, in which enhanced accuracy is enabled. The new technique is unconditionally stable (according to its parameter selection), it has no amplitude decay or overshooting, and it provides reduced period elongation errors. The method is based on displacement-velocity relations, requiring no computation of accelerations. It is efficient, simple and very easy to implement. Numerical results are presented along the paper, illustrating the good performance of the proposed technique. As it is described here, the new method has no drawbacks when compared to the Trapezoidal Rule (TR), which is one of… More >

Chih-Wen Chang¹, Chein-Shan Liu²

CMES-Computer Modeling in Engineering & Sciences, Vol.51, No.3, pp. 261-276, 2009, DOI:10.3970/cmes.2009.051.261

Abstract The backward advection-dispersion equation (ADE) for identifying the groundwater pollution source identification problems (GPSIPs) is numerically solved by employing a fictitious time integration method (FTIM). The backward ADE is renowned as ill-posed because the solution does not continuously count on the data. We transform the original parabolic equation into another parabolic type evolution equation by introducing a fictitious time coordinate, and adding a viscous damping coefficient to enhance the stability of numerical integration of the discretized equations by employing a group preserving scheme. When several numerical examples are amenable, we find that the FTIM is applicable to retrieve all past… More >

Chein-Shan Liu¹, Satya N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.36, No.3, pp. 261-286, 2008, DOI:10.3970/cmes.2008.036.261

Abstract The inverse Sturm-Liouville problem finds its applications in the identification of mechanical properties and/or geometrical configurations of a vibrating continuous medium; however, this problem is hard to solve, either theoretically or numerically. Previously, Liu (2008a) has constructed a Lie-group shooting method to determine the eigenvalues, and the corresponding eigenfunctions, for the direct Sturm-Liouville problem. In this study, we are concerned with solving the inverse Sturm-Liouville problem, by developing a Lie-group of SL(2,R) to construct nonlinear algebraic equations (NAEs), when discrete eigenvalues are specified. Our purpose here is to use these NAEs to solve the unknown function in the Sturm-Liouville operator.… More >

Chein-Shan Liu¹, Satya N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.34, No.2, pp. 155-178, 2008, DOI:10.3970/cmes.2008.034.155

Abstract In this paper we propose a novel method for solving a nonlinear optimization problem (NOP) under multiple equality and inequality constraints. The Kuhn-Tucker optimality conditions are used to transform the NOP into a mixed complementarity problem (MCP). With the aid of (nonlinear complementarity problem) NCP-functions a set of nonlinear algebraic equations is obtained. Then we develop a fictitious time integration method to solve these nonlinear equations. Several numerical examples of optimization problems, the inverse Cauchy problems and plasticity equations are used to demonstrate that the FTIM is highly efficient to calculate the NOPs and MCPs. The present method has some… More >

C.E. Majorana¹, V.A. Salomoni

CMES-Computer Modeling in Engineering & Sciences, Vol.33, No.1, pp. 49-84, 2008, DOI:10.3970/cmes.2008.033.049

Abstract A dynamic analysis of a thin shell finite element undergoing large displacements and rotations is here presented. The constitutive model adopted derives from the coupling of an hyperelastic basic model fulfilling a De Saint Venant-Kirchhoff criterion with a scalar damage function depending on the maximum value of a suitable strain measure attained through the deformation history; then plastic effects are included using an isotropic/kinematic hardening law. A conservative time integration scheme for the non-linear dynamics of the hyperelastic damaged-plastic thin shell is applied. The main characteristic of the scheme is to be conservative, since it allows for the time-discrete system… More >

Chein-Shan Liu^1,2, Hong-Hua Dai¹, Satya N. Atluri¹

CMES-Computer Modeling in Engineering & Sciences, Vol.81, No.2, pp. 195-228, 2011, DOI:10.3970/cmes.2011.081.195

Abstract In this continuation of a series of our earlier papers, we define a hyper-surface h(x,t) = 0 in terms of the unknown vector x, and a monotonically increasing function Q(t) of a time-like variable t, to solve a system of nonlinear algebraic equations F(x) = 0. If R is a vector related to ∂h / ∂x, , we consider the evolution equation x^· = λ[αR + βP], where P = F − R(F·R) / ||R||² such that P·R = 0; or x^· = λ[αF + βP^∗], where P^∗ = R − F(F·R) / ||F||² such that P^*·F =… More >

Sushil Mohan Ratnaker¹, Sanjay Mittal^1,2

CMES-Computer Modeling in Engineering & Sciences, Vol.80, No.3&4, pp. 179-200, 2011, DOI:10.3970/cmes.2011.080.179

Abstract Flow past a circular cylinder looses stability at Re ~ 47 via Hopf bifurcation. The eigenmode responsible for the instability leads to the von Kármán vortex shedding. In this work the linear stability of the flow to other modes, near the critical Re, is investigated. In particular, the study explores the possibility of modes other than the Kármán mode having the largest growth rate for Re < Re_cr. To this extent, global linear stability analysis (LSA) of the steady flow past a circular cylinder is carried out for Re = 45 and 48. In addition to the Kármán modes, two… More >

Chein-Shan Liu¹, Satya N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.71, No.3, pp. 279-304, 2011, DOI:10.3970/cmes.2011.071.279

Abstract For solving a system of nonlinear algebraic equations (NAEs) of the type: F(x)=0, or F_i(x_j) = 0, i,j = 1,...,n, a Newton-like algorithm has several drawbacks such as local convergence, being sensitive to the initial guess of solution, and the time-penalty involved in finding the inversion of the Jacobian matrix ∂F_i/∂x_j. Based-on an invariant manifold defined in the space of (x,t) in terms of the residual-norm of the vector F(x), we can derive a gradient-flow system of nonlinear ordinary differential equations (ODEs) governing the evolution of x with a fictitious time-like variable t as an independent variable. We can prove… More >