Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.13, No.2, pp. 149-164, 2006, DOI:10.3970/cmes.2006.013.149

Abstract The present paper provides a Lie-group shooting method for the numerical solutions of second order nonlinear boundary value problems exhibiting multiple solutions. It aims to find all solutions as easy as possible. The boundary conditions considered are classified into four types, namely the Dirichlet, the first Robin, the second Robin and the Neumann. The two Robin type problems are transformed into a canonical one by using the technique of symmetric extension of the governing equations. The Lie-group shooting method is very effective to search unknown initial condition through a weighting factor r ∈ (0,1) Furthermore, the More >

S.Y. Wang^1,2, M.Y. Wang³

CMES-Computer Modeling in Engineering & Sciences, Vol.13, No.2, pp. 119-148, 2006, DOI:10.3970/cmes.2006.013.119

Abstract In this paper, an implicit free boundary parametrization method is presented as an effective approach for simultaneous shape and topology optimization of structures. The moving free boundary of a structure is embedded as a zero level set of a higher dimensional implicit level set function. The radial basis functions (RBFs) are introduced to parametrize the implicit function with a high level of accuracy and smoothness. The motion of the free boundary is thus governed by a mathematically more convenient ordinary differential equation (ODE). Eigenvalue stability can be guaranteed due to the use of inverse multiquadric… More >

J. Sladek¹, V. Sladek¹, P. H. Wen², M.H. Aliabadi³

CMES-Computer Modeling in Engineering & Sciences, Vol.13, No.2, pp. 103-118, 2006, DOI:10.3970/cmes.2006.013.103

Abstract A meshless local Petrov-Galerkin (MLPG) method is applied to solve bending problems of shear deformable shallow shells described by the Reissner theory. Both static and dynamic loads are considered. For transient elastodynamic case the Laplace-transform is used to eliminate the time dependence of the field variables. A weak formulation with a unit test function transforms the set of governing equations into local integral equations on local subdomains in the mean surface of the shell. Nodal points are randomly spread on that surface and each node is surrounded by a circular subdomain to which local integral More >

Radek Pecher¹, Steve Elston, Peter Raynes

CMES-Computer Modeling in Engineering & Sciences, Vol.13, No.2, pp. 91-102, 2006, DOI:10.3970/cmes.2006.013.091

Abstract Meshfree techniques for solving partial differential equations in physics and engineering are a powerful new alternative to the traditional mesh-based techniques, such as the finite difference method or the finite element method. The elimination of the domain mesh enables, among other benefits, more efficient solutions of nonlinear and multi-scale problems. One particular example of these kinds of problems is a Q-tensor based model of nematic liquid crystals involving topological defects.
This paper presents the first application of the meshless local Petrov-Galerkin method to solving the Q-tensor equations of nematostatics. The theoretical part introduces the Landau More >

Zai You Yan¹

CMES-Computer Modeling in Engineering & Sciences, Vol.13, No.2, pp. 81-90, 2006, DOI:10.3970/cmes.2006.013.081

Abstract Boundary element method in acoustics for Neumann boundary condition problems including sharp edges & corners is investigated. In previous acoustic boundary element method, acoustic pressure and normal velocity are the two variables at sharp edges & corners. However, the normal velocity at sharp edges & corners is discontinuous due to the indefinite normal vector. To avoid the indefinite normal vector and the discontinuous normal velocity at sharp edges & corners, normal vector of elemental node is defined and applied in the numerical implementation. Then the normal velocity is transformed to velocity which is unique even More >

D. H. Pahr¹, F.G. Rammerstorfer¹

CMES-Computer Modeling in Engineering & Sciences, Vol.12, No.3, pp. 229-242, 2006, DOI:10.3970/cmes.2006.012.229

Abstract Sandwich structures are efficient lightweight materials. Due to there design they exhibit very special failure modes such as global buckling, shear crimping, facesheet wrinkling, facesheet dimpling, and face/core yielding. The core of the sandwich is usually made of foams or cellular materials, e.g., honeycombs. Especially in the case of honeycomb cores the correlation between analytical buckling predictions and experiments might be poor (Ley, Lin, and Uy (1999)). The reason for this lies in the fact that analytical formulae typically assume a homogeneous core (continuous support of the facesheets). This work highlights problems of honeycomb core… More >

Jin Ma, Hongbing Lu, Ranga Komanduri¹

CMES-Computer Modeling in Engineering & Sciences, Vol.12, No.3, pp. 213-228, 2006, DOI:10.3970/cmes.2006.012.213

Abstract The generalized interpolation material point (GIMP) method, recently developed using a C¹ continuous weighting function, has solved the numerical noise problem associated with material points just crossing the cell borders, so that it is suitable for simulation of relatively large deformation problems. However, this method typically uses a uniform mesh in computation when one level of material points is used, thus limiting its effectiveness in dealing with structures involving areas of high stress gradients. In this paper, a spatial refinement scheme of the structured grid for GIMP is presented for simulations with highly localized stress gradients.… More >

Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.12, No.3, pp. 197-212, 2006, DOI:10.3970/cmes.2006.012.197

Abstract In this paper we numerically solve the Burgers equation by semi-discretizing it at the n interior spatial grid points into a set of ordinary differential equations: u^· = f(u,t), u ∈ Rⁿ. Then, we take the dissipative behavior of Burgers equation into account by considering the magnitude ||u|| as another component; hence, an augmented quasilinear differential equations system X^˙ = AX with X := (u^T,||u||)^T ∈ Mⁿ⁺¹ is derived. According to a Lie algebra property of A∈so(n,1) we thus develop a new numerical scheme with the transformation matrix G∈SO_o(n,1) being an element of the proper orthochronous Lorentz group. More >

L. Gao¹, K. Liu^1,2, Y. Liu³

CMES-Computer Modeling in Engineering & Sciences, Vol.12, No.3, pp. 181-196, 2006, DOI:10.3970/cmes.2006.012.181

Abstract A new numerical algorithm based on the Meshless Local Petrov-Galerkin approach is presented for analyzing the dynamic fracture problems in elastic media. To simplify the treatment of essential boundary condition, a novel modified Moving Least Square (MLS) procedure is proposed by introducing Lagrange multiplier into MLS procedure, which can perform both MLS approximation and interpolation in one approximation domain. The compact spline function is used as the test function in the local form of elasto-dynamic equations. For the feature of stress wave propagation, the coupled second-order ODEs respect to the time are solved by the… More >

Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.12, No.3, pp. 161-180, 2006, DOI:10.3970/cmes.2006.012.161

Abstract Materials' internal spacetime may bear certain similarities with the external spacetime of special relativity theory. Previously, it is shown that material hardening and anisotropy may cause the internal spacetime curved. In this paper we announce the third mechanism of mixed-control to cause the curvedness of internal spacetime. To tackle the mixed-control problem for a Prager kinematic hardening material, we demonstrate two new formulations. By using two-integrating factors idea we can derive two Lie type systems in the product space of M^m+1⊗Mⁿ⁺¹. The Lie algebra is a direct sum of so(m,1)⊕so(n,1), and correspondingly the symmetry group is… More >