Trefftz solutions for piezoelectricity by Lekhnitskii's formalism and boundary‐collocation methodSheng, N.; Sze, K. Y.; Cheung, Y. K.
doi: 10.1002/nme.1523pmid: N/A
In this paper, a solution procedure for plane piezoelectricity is developed by Trefftz boundary‐collocation method. Starting with the general plane piezoelectricity solution derived by Lekhnitskii's formalism, the basic sets of Trefftz functions which satisfy the homogeneous governing equations are derived. Moreover, special sets of Trefftz functions are derived for impermeable elliptical voids, impermeable sharp cracks and permeable sharp cracks with arbitrary orientations with respect to the material poling direction. The functions in the special sets satisfy not only the homogeneous governing equations but also the boundary conditions at the peripheries of the pertinent defects. By adopting Trefftz functions as the trial functions, multi‐domain Trefftz boundary‐collocation method is formulated. Numerical examples are presented to illustrate the efficacy of the formulation. Copyright © 2005 John Wiley & Sons, Ltd.
Robust adaptive remeshing strategy for large deformation, transient impact simulationsErhart, Tobias; Wall, Wolfgang A.; Ramm, Ekkehard
doi: 10.1002/nme.1531pmid: N/A
In this paper, an adaptive approach, with remeshing as essential ingredient, towards robust and efficient simulation techniques for fast transient, highly non‐linear processes including contact is discussed. The necessity for remeshing stems from two sources: the capability to deal with large deformations that might even require topological changes of the mesh and the desire for an error driven distribution of computational resources. The overall computational approach is sketched, the adaptive remeshing strategy is presented and the crucial aspect, the choice of suitable error indicator(s), is discussed in more detail. Several numerical examples demonstrate the performance of the approach. Copyright © 2005 John Wiley & Sons, Ltd.
Local maximum‐entropy approximation schemes: a seamless bridge between finite elements and meshfree methodsArroyo, M.; Ortiz, M.
doi: 10.1002/nme.1534pmid: N/A
We present a one‐parameter family of approximation schemes, which we refer to as local maximum‐entropy approximation schemes, that bridges continuously two important limits: Delaunay triangulation and maximum‐entropy (max‐ent) statistical inference. Local max‐ent approximation schemes represent a compromise—in the sense of Pareto optimality—between the competing objectives of unbiased statistical inference from the nodal data and the definition of local shape functions of least width. Local max‐ent approximation schemes are entirely defined by the node set and the domain of analysis, and the shape functions are positive, interpolate affine functions exactly, and have a weak Kronecker‐delta property at the boundary. Local max‐ent approximation may be regarded as a regularization, or thermalization, of Delaunay triangulation which effectively resolves the degenerate cases resulting from the lack or uniqueness of the triangulation. Local max‐ent approximation schemes can be taken as a convenient basis for the numerical solution of PDEs in the style of meshfree Galerkin methods. In test cases characterized by smooth solutions we find that the accuracy of local max‐ent approximation schemes is vastly superior to that of finite elements. Copyright © 2005 John Wiley & Sons, Ltd.
A method for modal reanalysis of topological modifications of structuresYang, Zhi Jun; Chen, Su Huan; Wu, Xiao Ming
doi: 10.1002/nme.1546pmid: N/A
A method for structural modal reanalysis for three cases of topological modifications, the number of degrees of freedom (DOFs) is unchanged, decreased, and increased, is presented. In this method, the newly added DOFs are linked to the original DOFs of the modified structure by means of the dynamic reduction so as to obtain the condensed equation. The methods for forming the stiffness and mass increments, ΔK and ΔM, resulting from the three cases of topological modifications of structures are discussed. The extended Kirsch method is used to improve the accuracy of the starting solutions of the initial structure. And then, the eigenvectors of newly added DOFs resulting from topological modification can be recovered. At last, the Rayleigh–Ritz analysis is used to evaluate the eigenvalues and eigenvectors for the modified structure. Three numerical examples are given to illustrate the applications of the present approach. The results show that the proposed method is effective for structural modal reanalysis of three cases of the topological modifications and it is easy to implement on a computer. By comparing with previous method, it can be seen that the present method can give good approximate eigenvalues and eigenvectors, even if the topological modifications are very large. Copyright © 2005 John Wiley & Sons, Ltd.
Crush dynamics of square honeycomb sandwich coresXue, Zhenyu; Hutchinson, John W.
doi: 10.1002/nme.1535pmid: N/A
Square honeycombs are effective as cores for all‐metal sandwich plates in that they combine excellent crushing strength and energy absorption with good stiffness and strength in out‐of‐plane shear and in‐plane stretch. In applications where sandwich plates must absorb significant energy in crushing under intense impulsive loads, dynamic effects play a significant role in the behaviour of the core. Three distinct dynamic effects can be identified: (i) inertial resistance, (ii) inertial stabilization of webs against buckling, and (iii) material strain‐rate dependence. Each contributes to dynamic strengthening of the core. These effects are illustrated and quantified with the aid of detailed numerical calculations for rates of deformation characteristic of shock loads in air and water. A continuum model for high rate deformation of square honeycomb cores is introduced that can be used to simulate core behaviour in large structural calculations when it is not feasible to mesh the detailed core geometry. The performance of the continuum model is demonstrated for crushing deformations. Copyright © 2005 John Wiley & Sons, Ltd.
A hybrid regularization method for inverse heat conduction problemsLing, Xianwu; Cherukuri, H. P.; Horstemeyer, M. F.
doi: 10.1002/nme.1540pmid: N/A
This paper presents a hybrid regularization method for solving inverse heat conduction problems. The method uses future temperatures and past fluxes to reduce the sensitivity to temperature noise. A straightforward comparison technique is suggested to find the optimal number of the future temperatures. Also, an eigenvalue reduction technique is used to further improve the accuracy of the inverse solution. The method provides a physical insight into the inverse problems under study. The insight indicates that the inverse algorithm is a general purpose algorithm and applicable to various numerical methods (although our development was based on FEM), and that the inverse solutions can be obtained by directly extending Stolz's equation in the least‐squares error (LSE) sense. Direct extension of the present method to the inverse internal heat generation problems is made. Four numerical examples are given to validate the method. The effects of the future temperatures, the past fluxes, the eigenvalue reduction, the varying number of future temperatures and local iterations for non‐linear problems are studied. Copyright © 2005 John Wiley & Sons, Ltd.
The development of hybrid axisymmetric elements based on the Hellinger–Reissner variational principleJog, C. S.; Annabattula, R.
doi: 10.1002/nme.1552pmid: N/A
We present a general procedure for the development of hybrid axisymmetric elements based on the Hellinger–Reissner principle within the context of linear elasticity. Similar to planar elements, the stress interpolation is obtained by an identification of the zero‐energy modes. We illustrate our procedure by designing a lower‐order (four‐node) and a higher‐order (nine‐node) element. Both elements are of correct rank, and moreover use the minimum number of stress parameters, namely seven and 17. Several examples are presented to show the excellent performance of both elements under various demanding situations such as when the material is almost incompressible, when the thickness to radius ratio is very small, etc. When the variation of the field variables is along the radial direction alone, when the mesh is uniform, and the loading is of pressure type, the developed elements are superconvergent, i.e. they yield the exact nodal displacement values. Copyright © 2005 John Wiley & Sons, Ltd.