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Unbounded Domains (unbounded + domain)
Selected AbstractsSemi-analytical elastostatic analysis of unbounded two-dimensional domainsINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 11 2002Andrew J. Deeks Abstract Unbounded plane stress and plane strain domains subjected to static loading undergo infinite displacements, even when the zero displacement boundary condition at infinity is enforced. However, the stress and strain fields are well behaved, and are of practical interest. This causes significant difficulty when analysis is attempted using displacement-based numerical methods, such as the finite-element method. To circumvent this difficulty problems of this nature are often changed subtly before analysis to limit the displacements to finite values. Such a process is unsatisfactory, as it distorts the solution in some way, and may lead to a stiffness matrix that is nearly singular. In this paper, the semi-analytical scaled boundary finite-element method is extended to permit the analysis of such problems without requiring any modification of the problem itself. This is possible because the governing differential equations are solved analytically in the radial direction. The displacement solutions so obtained include an infinite component, but relative motion between any two points in the unbounded domain can be computed accurately. No small arbitrary constants are introduced, no arbitrary truncation of the domain is performed, and no ill-conditioned matrices are inverted. Copyright © 2002 John Wiley & Sons, Ltd. [source] The control-theory-based artificial boundary conditions for time-dependent wave guide problems in unbounded domainINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 12 2005Tianyun Liu Abstract A method is proposed to obtain the high-performance artificial boundary conditions for solving the time-dependent wave guide problems in an unbounded domain. Using the variable separation method, it is possible to reduce the spatial variables of the wave equation by one. Furthermore, introducing auxiliary functions makes the reduced wave equation a linear first-order ordinary differential system with one control input. Solving the closed-loop control system, a stable and accurate artificial boundary condition is obtained in a rigorous mathematical manner. Numerical examples have demonstrated the effectiveness of the proposed artificial boundary conditions for the time-dependent wave guide problems in unbounded domain. Copyright © 2005 John Wiley & Sons, Ltd. [source] A continued-fraction-based high-order transmitting boundary for wave propagation in unbounded domains of arbitrary geometryINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2008Mohammad Hossein Bazyar Abstract A high-order local transmitting boundary is developed to model the propagation of elastic waves in unbounded domains. This transmitting boundary is applicable to scalar and vector waves, to unbounded domains of arbitrary geometry and to anisotropic materials. The formulation is based on a continued-fraction solution of the dynamic-stiffness matrix of an unbounded domain. The coefficient matrices of the continued fraction are determined recursively from the scaled boundary finite element equation in dynamic stiffness. The solution converges rapidly over the whole frequency range as the order of the continued fraction increases. Using the continued-fraction solution and introducing auxiliary variables, a high-order local transmitting boundary is formulated as an equation of motion with symmetric and frequency-independent coefficient matrices. It can be coupled seamlessly with finite elements. Standard procedures in structural dynamics are directly applicable for evaluating the response in the frequency and time domains. Analytical and numerical examples demonstrate the high rate of convergence and efficiency of this high-order local transmitting boundary. Copyright © 2007 John Wiley & Sons, Ltd. [source] Perfectly matched layers for transient elastodynamics of unbounded domainsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2004Ushnish Basu Abstract One approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outward from the bounded domain. A perfectly matched layer (PML) is an unphysical absorbing layer model for linear wave equations that absorbs, almost perfectly, outgoing waves of all non-tangential angles-of-incidence and of all non-zero frequencies. In a recent work [Computer Methods in Applied Mechanics and Engineering 2003; 192: 1337,1375], the authors presented, inter alia, time-harmonic governing equations of PMLs for anti-plane and for plane-strain motion of (visco-) elastic media. This paper presents (a) corresponding time-domain, displacement-based governing equations of these PMLs and (b) displacement-based finite element implementations of these equations, suitable for direct transient analysis. The finite element implementation of the anti-plane PML is found to be symmetric, whereas that of the plane-strain PML is not. Numerical results are presented for the anti-plane motion of a semi-infinite layer on a rigid base, and for the classical soil,structure interaction problems of a rigid strip-footing on (i) a half-plane, (ii) a layer on a half-plane, and (iii) a layer on a rigid base. These results demonstrate the high accuracy achievable by PML models even with small bounded domains. Copyright © 2004 John Wiley & Sons, Ltd. [source] Time asymptotics for the polyharmonic wave equation in waveguidesMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2003P. H. Lesky Abstract Let , denote an unbounded domain in ,n having the form ,=,l×D with bounded cross-section D,,n,l, and let m,, be fixed. This article considers solutions u to the scalar wave equation ,u(t,x) +(,,)mu(t,x) = f(x)e,i,t satisfying the homogeneous Dirichlet boundary condition. The asymptotic behaviour of u as t,, is investigated. Depending on the choice of f ,, and ,, two cases occur: Either u shows resonance, which means that ,u(t,x),,, as t,, for almost every x , ,, or u satisfies the principle of limiting amplitude. Furthermore, the resolvent of the spatial operators and the validity of the principle of limiting absorption are studied. Copyright © 2003 John Wiley & Sons, Ltd. [source] Numerical solution to a linearized KdV equation on unbounded domainNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2008Chunxiong Zheng Abstract Exact absorbing boundary conditions for a linearized KdV equation are derived in this paper. Applying these boundary conditions at artificial boundary points yields an initial-boundary value problem defined only on a finite interval. A dual-Petrov-Galerkin scheme is proposed for numerical approximation. Fast evaluation method is developed to deal with convolutions involved in the exact absorbing boundary conditions. In the end, some numerical tests are presented to demonstrate the effectiveness and efficiency of the proposed method.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008 [source] Local existence for the free boundary problem for nonrelativistic and Relativistic compressible Euler equations with a vacuum boundary conditionCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 11 2009Yuri Trakhinin We study the free boundary problem for the equations of compressible Euler equations with a vacuum boundary condition. Our main goal is to recover in Eulerian coordinates the earlier well-posedness result obtained by Lindblad [11] for the isentropic Euler equations and extend it to the case of full gas dynamics. For technical simplicity we consider the case of an unbounded domain whose boundary has the form of a graph and make short comments about the case of a bounded domain. We prove the local-in-time existence in Sobolev spaces by the technique applied earlier to weakly stable shock waves and characteristic discontinuities [5, 12]. It contains, in particular, the reduction to a fixed domain, using the "good unknown" of Alinhac [1], and a suitable Nash-Moser-type iteration scheme. A certain modification of such an approach is caused by the fact that the symbol associated to the free surface is not elliptic. This approach is still directly applicable to the relativistic version of our problem in the setting of special relativity, and we briefly discuss its extension to general relativity. © 2009 Wiley Periodicals, Inc. [source] The attractor for a nonlinear reaction-diffusion system in an unbounded domainCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 6 2001Messoud A. Efendiev In this paper the quasi-linear second-order parabolic systems of reaction-diffusion type in an unbounded domain are considered. Our aim is to study the long-time behavior of parabolic systems for which the nonlinearity depends explicitly on the gradient of the unknown functions. To this end we give a systematic study of given parabolic systems and their attractors in weighted Sobolev spaces. Dependence of the Hausdorff dimension of attractors on the weight of the Sobolev spaces is considered. © 2001 John Wiley & Sons, Inc. [source] Semi-analytical far field model for three-dimensional finite-element analysisINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 11 2004James P. Doherty Abstract A challenging computational problem arises when a discrete structure (e.g. foundation) interacts with an unbounded medium (e.g. deep soil deposit), particularly if general loading conditions and non-linear material behaviour is assumed. In this paper, a novel method for dealing with such a problem is formulated by combining conventional three-dimensional finite-elements with the recently developed scaled boundary finite-element method. The scaled boundary finite-element method is a semi-analytical technique based on finite-elements that obtains a symmetric stiffness matrix with respect to degrees of freedom on a discretized boundary. The method is particularly well suited to modelling unbounded domains as analytical solutions are found in a radial co-ordinate direction, but, unlike the boundary-element method, no complex fundamental solution is required. A technique for coupling the stiffness matrix of bounded three-dimensional finite-element domain with the stiffness matrix of the unbounded scaled boundary finite-element domain, which uses a Fourier series to model the variation of displacement in the circumferential direction of the cylindrical co-ordinate system, is described. The accuracy and computational efficiency of the new formulation is demonstrated through the linear elastic analysis of rigid circular and square footings. Copyright © 2004 John Wiley & Sons, Ltd. [source] Coupling of mapped wave infinite elements and plane wave basis finite elements for the Helmholtz equation in exterior domainsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2003Rie Sugimoto Abstract The theory for coupling of mapped wave infinite elements and special wave finite elements for the solution of the Helmholtz equation in unbounded domains is presented. Mapped wave infinite elements can be applied to boundaries of arbitrary shape for exterior wave problems without truncation of the domain. Special wave finite elements allow an element to contain many wavelengths rather than having many finite elements per wavelength like conventional finite elements. Both types of elements include trigonometric functions to describe wave behaviour in their shape functions. However the wave directions between nodes on the finite element/infinite element interface can be incompatible. This is because the directions are normally globally constant within a special finite element but are usually radial from the ,pole' within a mapped wave infinite element. Therefore forcing the waves associated with nodes on the interface to be strictly radial is necessary to eliminate this internode incompatibility. The coupling of these elements was tested for a Hankel source problem and plane wave scattering by a cylinder and good accuracy was achieved. This paper deals with unconjugated infinite elements and is restricted to two-dimensional problems. Copyright © 2003 John Wiley & Sons, Ltd. [source] A continued-fraction-based high-order transmitting boundary for wave propagation in unbounded domains of arbitrary geometryINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2008Mohammad Hossein Bazyar Abstract A high-order local transmitting boundary is developed to model the propagation of elastic waves in unbounded domains. This transmitting boundary is applicable to scalar and vector waves, to unbounded domains of arbitrary geometry and to anisotropic materials. The formulation is based on a continued-fraction solution of the dynamic-stiffness matrix of an unbounded domain. The coefficient matrices of the continued fraction are determined recursively from the scaled boundary finite element equation in dynamic stiffness. The solution converges rapidly over the whole frequency range as the order of the continued fraction increases. Using the continued-fraction solution and introducing auxiliary variables, a high-order local transmitting boundary is formulated as an equation of motion with symmetric and frequency-independent coefficient matrices. It can be coupled seamlessly with finite elements. Standard procedures in structural dynamics are directly applicable for evaluating the response in the frequency and time domains. Analytical and numerical examples demonstrate the high rate of convergence and efficiency of this high-order local transmitting boundary. Copyright © 2007 John Wiley & Sons, Ltd. [source] A non-reflecting layer method for non-linear wave-type equations on unbounded domains with applications to shape memory alloy rodsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 15 2005M. Newman Abstract In this paper a new technique is introduced and applied in solving one-dimensional linear and non-linear wave-type equations on an unbounded spatial domain. This new technique referred to as the non-reflecting layer method (NRLM) extends the computational domain with an artificial layer on which a one-way wave equation is solved. The method will be applied to compute stress waves in long rods consisting of NiTi shape memory alloy material subjected to impact loading and undergoing detwinning and pseudo-elastic material responses. The NRLM has been tested on model problems and it has been found that the computed solutions agree well with the exact solutions, i.e. normalized error levels are in ranges acceptable for engineering computations. Copyright © 2005 John Wiley & Sons, Ltd. [source] Improved accuracy for the Helmholtz equation in unbounded domainsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 15 2004Eli Turkel Abstract Based on properties of the Helmholtz equation, we derive a new equation for an auxiliary variable. This reduces much of the oscillations of the solution leading to more accurate numerical approximations to the original unknown. Computations confirm the improved accuracy of the new models in both two and three dimensions. This also improves the accuracy when one wants the solution at neighbouring wavenumbers by using an expansion in k. We examine the accuracy for both waveguide and scattering problems as a function of k, h and the forcing mode l. The use of local absorbing boundary conditions is also examined as well as the location of the outer surface as functions of k. Connections with parabolic approximations are analysed. Copyright © 2004 John Wiley & Sons, Ltd. [source] Perfectly matched layers for transient elastodynamics of unbounded domainsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2004Ushnish Basu Abstract One approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outward from the bounded domain. A perfectly matched layer (PML) is an unphysical absorbing layer model for linear wave equations that absorbs, almost perfectly, outgoing waves of all non-tangential angles-of-incidence and of all non-zero frequencies. In a recent work [Computer Methods in Applied Mechanics and Engineering 2003; 192: 1337,1375], the authors presented, inter alia, time-harmonic governing equations of PMLs for anti-plane and for plane-strain motion of (visco-) elastic media. This paper presents (a) corresponding time-domain, displacement-based governing equations of these PMLs and (b) displacement-based finite element implementations of these equations, suitable for direct transient analysis. The finite element implementation of the anti-plane PML is found to be symmetric, whereas that of the plane-strain PML is not. Numerical results are presented for the anti-plane motion of a semi-infinite layer on a rigid base, and for the classical soil,structure interaction problems of a rigid strip-footing on (i) a half-plane, (ii) a layer on a half-plane, and (iii) a layer on a rigid base. These results demonstrate the high accuracy achievable by PML models even with small bounded domains. Copyright © 2004 John Wiley & Sons, Ltd. [source] High-order boundary conditions for linearized shallow water equations with stratification, dispersion and advection,INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2004Vince J. van Joolen Abstract The two-dimensional linearized shallow water equations are considered in unbounded domains with density stratification. Wave dispersion and advection effects are also taken into account. The infinite domain is truncated via a rectangular artificial boundary ,, and a high-order open boundary condition (OBC) is imposed on ,. Then the problem is solved numerically in the finite domain bounded by ,. A recently developed boundary scheme is employed, which is based on a reformulation of the sequence of OBCs originally proposed by Higdon. The OBCs can easily be used up to any desired order. They are incorporated here in a finite difference scheme. Numerical examples are used to demonstrate the performance and advantages of the computational method, with an emphasis is on the effect of stratification. Published in 2004 by John Wiley & Sons, Ltd. [source] Potential flow around obstacles using the scaled boundary finite-element methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2003Andrew J. Deeks Abstract The scaled boundary finite-element method is a novel semi-analytical technique, combining the advantages of the finite element and the boundary element methods with unique properties of its own. The method works by weakening the governing differential equations in one co-ordinate direction through the introduction of shape functions, then solving the weakened equations analytically in the other (radial) co-ordinate direction. These co-ordinate directions are defined by the geometry of the domain and a scaling centre. The method can be employed for both bounded and unbounded domains. This paper applies the method to problems of potential flow around streamlined and bluff obstacles in an infinite domain. The method is derived using a weighted residual approach and extended to include the necessary velocity boundary conditions at infinity. The ability of the method to model unbounded problems is demonstrated, together with its ability to model singular points in the near field in the case of bluff obstacles. Flow fields around circular and square cylinders are computed, graphically illustrating the accuracy of the technique, and two further practical examples are also presented. Comparisons are made with boundary element and finite difference solutions. Copyright © 2003 John Wiley & Sons, Ltd. [source] The asymptotic behaviour of weak solutions to the forward problem of electrical impedance tomography on unbounded three-dimensional domainsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 2 2009Michael Lukaschewitsch Abstract The forward problem of electrical impedance tomography on unbounded domains can be studied by introducing appropriate function spaces for this setting. In this paper we derive the point-wise asymptotic behaviour of weak solutions to this problem in the three-dimensional case. Copyright © 2008 John Wiley & Sons, Ltd. [source] On the asymptotic stability of steady solutions of the Navier,Stokes equations in unbounded domainsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2007Francesca Crispo Abstract We consider the problem of the asymptotic behaviour in the L2 -norm of solutions of the Navier,Stokes equations. We consider perturbations to the rest state and to stationary motions. In both cases we study the initial-boundary value problem in unbounded domains with non-compact boundary. In particular, we deal with domains with varying and possibly divergent exits to infinity and aperture domains. Copyright © 2007 John Wiley & Sons, Ltd. [source] Artificial boundary conditions for Petrovsky systems of second order in exterior domains and in other domains of conical typeMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2004A. Nazarov, Sergue Abstract The approximation of solutions to boundary value problems on unbounded domains by those on bounded domains is one of the main applications for artificial boundary conditions. Based on asymptotic analysis, here a new method is presented to construct local artificial boundary conditions for a very general class of elliptic problems where the main asymptotic term is not known explicitly. Existence and uniqueness of approximating solutions are proved together with asymptotically precise error estimates. One class of important examples includes boundary value problems for anisotropic elasticity and piezoelectricity. Copyright © 2004 John Wiley & Sons, Ltd. [source] The diffraction in a class of unbounded domains connected through a holeMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2003Yu. V. Shestopalov Abstract In this paper, the unique solvability, Fredholm property, and the principle of limiting absorption are proved for a boundary value problem for the system of Maxwell's equations in a semi-infinite rectangular cylinder coupled with a layer by an aperture of arbitrary shape. Conditions at infinity are taken in the form of the Sveshnikov,Werner partial radiation conditions. The method of solution employs Green's functions of the partial domains and reduction to vector pseudodifferential equations considered in appropriate vectorial Sobolev spaces. Singularities of Green's functions are separated both in the domain and on its boundary. The smoothness of solutions is established. Copyright © 2003 John Wiley & Sons, Ltd. [source] Elliptic and parabolic problems in unbounded domainsMATHEMATISCHE NACHRICHTEN, Issue 1 2004Patrick Guidotti Abstract We consider elliptic and parabolic problems in unbounded domains. We give general existence and regularity results in Besov spaces and semi-explicit representation formulas via operator-valued fundamental solutions which turn out to be a powerful tool to derive a series of qualitative results about the solutions. We give a sample of possible applications including asymptotic behavior in the large, singular perturbations, exact boundary conditions on artificial boundaries and validity of maximum principles. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] A stabilized Hermite spectral method for second-order differential equations in unbounded domainsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2007Heping Ma Abstract A stabilized Hermite spectral method, which uses the Hermite polynomials as trial functions, is presented for the heat equation and the generalized Burgers equation in unbounded domains. In order to overcome instability that may occur in direct Hermite spectral methods, a time-dependent scaling factor is employed in the Hermite expansions. The stability of the scheme is examined and optimal error estimates are derived. Numerical experiments are given to confirm the theoretical results.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source] Large Scale Simulation with Scaled Boundary Finite Element MethodPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2009Marco Schauer Nowadays scientific and engineering applications often require wave propagation in infinite or unbounded domains. In order to model such applications we separate our model into near-field and far-field. The near-field is represented by the well-known finite element method (FEM), whereas the far-field is mapped by a scaled boundary finite element (SBFE) approach. This latter approach allows wave propagation in infinite domains and suppresses the reflection of waves at the boundary, thus being a suitable method to model wave propagation to infinity. It is non-local in time and space. From a computational point of view, those characteristics are a drawback because they lead to storage consuming calculations with high computational time-effort. The non-locality in space causes fully populated unit-impulse acceleration influence matrices for each time step, leading to immense storage consumption for problems with a large number of degrees of freedom. Additionally, a different influence matrix has to be assembled for each time step which yields unacceptable storage requirements for long simulation times. For long slender domains, where many nodes are rather far from each other and where the influence of the degrees of freedom of those distant nodes is neglectable, substructuring represents an efficient method to reduce storage requirements and computational effort. The presented simulation with substructuring still yields satisfactory results. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Fractional calculus applied to radiation dampingPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003C. Trinks Dipl.-Ing. Separate treatment of the low, and high,frequency part of the dynamic stiffness is essential when approximating the latter in the frequency,domain. In this paper, a doubly,asymptotic rational approximation of the low,frequency part is combined with an analytical interpretation of the asymptotic part leading to a system of fractional differential equations to represent the force,displacement relationship. Here, an analogy between fading memory of viscoelastic materials and radiation damping of unbounded domains is visible. [source] A representation of acoustic waves in unbounded domains,COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 10 2005Bradley K. Alpert Compact, time-harmonic, acoustic sources produce waves that decay too slowly to be square-integrable on a line away from the sources. We introduce an inner product, arising directly from Green's second theorem, to form a Hilbert space of these waves and present examples of its computation.1 © 2005 Wiley Periodicals, Inc. 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