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Normal Derivatives (normal + derivative)
Selected AbstractsA discontinuous enrichment method for the efficient solution of plate vibration problems in the medium-frequency regimeINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2010Paolo Massimi Abstract A discontinuous enrichment method (DEM) is presented for the efficient discretization of plate vibration problems in the medium-frequency regime. This method enriches the polynomial shape functions of the classical finite element discretization with free-space solutions of the biharmonic operator governing the elastic vibrations of an infinite Kirchhoff plate. These free-space solutions, which represent flexural waves and decaying modes, are discontinuous across the element interfaces. For this reason, two different and carefully constructed Lagrange multiplier approximations are introduced along the element edges to enforce a weak continuity of the transversal displacement and its normal derivative, and discrete Lagrange multipliers are introduced at the element corners to enforce there a weak continuity of the transversal displacement. The proposed DEM is illustrated with the solution of sample plate vibration problems with different types of harmonic loading in the medium-frequency regime, away from and close to resonance. In all cases, its performance is found to be significantly superior to that of the classical higher-order finite element method. Copyright © 2010 John Wiley & Sons, Ltd. [source] An advanced boundary element method for solving 2D and 3D static problems in Mindlin's strain-gradient theory of elasticityINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2010G. F. Karlis Abstract An advanced boundary element method (BEM) for solving two- (2D) and three-dimensional (3D) problems in materials with microstructural effects is presented. The analysis is performed in the context of Mindlin's Form-II gradient elastic theory. The fundamental solution of the equilibrium partial differential equation is explicitly derived. The integral representation of the problem, consisting of two boundary integral equations, one for displacements and the other for its normal derivative, is developed. The global boundary of the analyzed domain is discretized into quadratic line and quadrilateral elements for 2D and 3D problems, respectively. Representative 2D and 3D numerical examples are presented to illustrate the method, demonstrate its accuracy and efficiency and assess the gradient effect on the response. The importance of satisfying the correct boundary conditions in gradient elastic problems is illustrated with the solution of simple 2D problems. Copyright © 2010 John Wiley & Sons, Ltd. [source] A discontinuous Galerkin method for elliptic interface problems with application to electroporationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2009Grégory Guyomarc'h Abstract We solve elliptic interface problems using a discontinuous Galerkin (DG) method, for which discontinuities in the solution and in its normal derivatives are prescribed on an interface inside the domain. Standard ways to solve interface problems with finite element methods consist in enforcing the prescribed discontinuity of the solution in the finite element space. Here, we show that the DG method provides a natural framework to enforce both discontinuities weakly in the DG formulation, provided the triangulation of the domain is fitted to the interface. The resulting discretization leads to a symmetric system that can be efficiently solved with standard algorithms. The method is shown to be optimally convergent in the L2 -norm. We apply our method to the numerical study of electroporation, a widely used medical technique with applications to gene therapy and cancer treatment. Mathematical models of electroporation involve elliptic problems with dynamic interface conditions. We discretize such problems into a sequence of elliptic interface problems that can be solved by our method. We obtain numerical results that agree with known exact solutions. Copyright © 2008 John Wiley & Sons, Ltd. [source] A numerical method to solve the m -terms of a submerged body with forward speedINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 5 2002W.-Y. Duan Abstract To model mathematically the problem of a rigid body moving below the free surface, a control surface surrounding the body is introduced. The linear free surface condition of the steady waves created by the moving body is satisfied. To describe the fluid flow outside this surface a potential integral equation is constructed using the Kelvin wave Green function whereas inside the surface, a source integral equation is developed adopting a simple Green function. Source strengths are determined by matching the two integral equations through continuity conditions applied to velocity potential and its normal derivatives along the control surface. After solving for the induced fluid velocity on the body surface and the control surface, an integral equation is derived involving a mixed distribution of sources and dipoles using a simple Green function and one component of the fluid velocity. The normal derivatives of the fluid velocity on the body surface, namely the m -terms, are then solved by this matching integral equation method (MIEM). Numerical results are presented for two elliptical sections moving at a prescribed Froude number and submerged depth and a sensitivity analysis undertaken to assess the influence of these parameters. Furthermore, comparisons are performed to analyse the impact of different assumptions adopted in the derivation of the m -terms. It is found that the present method is easy to use in a panel method with satisfactory numerical precision. Copyright © 2002 John Wiley & Sons, Ltd. [source] On explicit solutions to the stationary axisymmetric Einstein-Maxwell equations describing dust disksANNALEN DER PHYSIK, Issue 10 2003C. Klein Abstract We review explicit solutions to the stationary axisymmetric Einstein-Maxwell equations which can be interpreted as disks of charged dust. The disks of finite or infinite extension are infinitesimally thin and constitute a surface layer at the boundary of an electro-vacuum. The Einstein-Maxwell equations in the presence of one Killing vector are obtained by using a projection formalism. This leads to equations for three-dimensional gravity where the matter is given by a SU(2,1)/S[U(1,1)× U(1)] nonlinear sigma model. The SU(2,1) invariance of the stationary Einstein-Maxwell equations can be used to construct solutions for the electro-vacuum from solutions to the pure vacuum case via a so-called Harrison transformation. It is shown that the corresponding solutions will always have a non-vanishing total charge and a gyromagnetic ratio of 2. Since the vacuum and the electro-vacuum equations in the stationary axisymmetric case are completely integrable, large classes of solutions can be constructed with techniques from the theory of solitons. The richest class of physically interesting solutions to the pure vacuum case due to Korotkin is given in terms of hyperelliptic theta functions. Harrison transformed hyperelliptic solutions are discussed. As a concrete example we study the transformation of a family of counter-rotating dust disks. To obtain algebro-geometric solutions with vanishing total charge which are of astrophysical relevance, three-sheeted surfaces have to be considered. The matter in the disk is discussed following Bi,ák et al. We review the ,cut and glue' technique where a strip is removed from an explicitly known spacetime and where the remainder is glued together after displacement. The discontinuities of the normal derivatives of the metric at the glueing hypersurface lead to infinite disks. If the energy conditions are satisfied and if the pressure is positive, the disks can be interpreted in the vacuum case as made up of two components of counter-rotating dust moving on geodesics. In electro-vacuum the condition of geodesic movement is replaced by electro-geodesic movement. As an example we discuss a class of Harrison-transformed hyperelliptic solutions. The range of parameters is identified where an interpretation of the matter in the disk in terms of electro-dust can be given. [source] | ||||||||||