Showing papers on "Biharmonic equation published in 2012"
Book•
17 Jan 2012
TL;DR: In this paper, the authors present an extension of the Dirichlet problem for the case of Perforated Domains with a Non-Periodic Structure, where the boundary value problem is solved with Neumann conditions on the outer part of the boundary and on the surface of the Cavities.
Abstract: Some Mathematical Problems of the Theory of Elasticity. Some Functional Spaces and Their Properties. Auxiliary Propositions. Korn's Inequalities. Boundary Value Problems of Linear Elasticity. Perforated Domains with a Periodic Structure. Extension Theorems. Estimates for Solutions of Boundary Value Problems of Elasticity in Perforated Domains. Periodic Solutions of Boundary Value Problems for the System of Elasticity. Saint-Venant's Principle for Periodic Solutions of the Elasticity System. Estimates and Existence Theorems for Solutions of the Elasticity System in Unbounded Domains. Strong G -Convergence of Elasticity Operators. Homogenization of the System of Linear Elasticity. Composites and Perforated Materials. The Mixed Problem in a Perforated Domain with the Dirichlet Boundary Conditions on the Outer Part of the Boundary and the Neumann Conditions on the Surface of the Cavities. The Boundary Value Problem with Neumann Conditions in a Perforated Domain. Asymptotic Expansions for Solutions of Boundary Value Problems of Elasticity in a Perforated Layer. Asymptotic Expansions for Solutions of the Dirichlet Problem for the Elasticity System in a Perforated Domain. Asymptotic Expansions for Solutions of the Dirichlet Problem for the Biharmonic Equation. Some Generalizations for the Case of Perforated Domains with a Non-Periodic Structure. Homogenization of the System of Elasticity with Almost-Periodic Coefficients. Homogenization of Stratified Structures. Estimates for the Rate of G -Convergence of Higher-Order Elliptic Operators. Spectral Problems . Some Theorems from Functional Analysis. Spectral Problems for Abstract Operators. Homogenization of Eigenvalues and Eigenfunctions of Boundary Value Problems for Strongly Non-Homogeneous Elastic Bodies. On the Behaviour of Eigenvalues and Eigenfunctions of the Dirichlet Problem for Second Order Elliptic Equations in Perforated Domains. Third Boundary Value Problem for Second Order Elliptic Equations in Domains with Rapidly Oscillating Boundary. Free Vibrations of Bodies with Concentrated Masses. On the Behaviour of Eigenvalues of the Dirchlet Problem in Domains with Cavities Whose Concentration is Small. Homogenization of Eigenvalues of Ordinary Differential Operators. Asymptotic Expansion of Eigenvalues and Eigenfunctions of the Sturm-Liouville Problem for Equations with Rapidly Oscillating Coefficients. On the Behaviour of the Eigenvalues and Eigenfunctions of a G -Convergent Sequence of Non-Self-Adjoint Operators. References.
838 citations
TL;DR: In this article, the generalized Chen's conjecture on biharmonic submanifolds is shown to be false by constructing foliations of proper bi-harmonic hyperplanes in a 5-dimensional conformally flat space with negative sectional curvature.
Abstract: The generalized Chen's conjecture on biharmonic submanifolds asserts that any biharmonic submanifold of a non-positively curved manifold is minimal (see e.g., [CMO1], [MO], [BMO1], [BMO2], [BMO3], [Ba1], [Ba2], [Ou1], [Ou2], [IIU]). In this paper, we prove that this conjecture is false by constructing foliations of proper biharmonic hyperplanes in a 5-dimensional conformally flat space with negative sectional curvature. Many examples of proper biharmonic submanifolds of non-positively curved spaces are also given.
92 citations
TL;DR: In this article, a variational method for numerical homogenization of divergence form elliptic, parabolic and hyperbolic equations with arbitrary rough coefficients was proposed. But this method does not rely on ergodicity or scale-separation but on compactness properties of the solution space and a new variational approach to homogenisation.
Abstract: We introduce a new variational method for the numerical homogenization of divergence form elliptic, parabolic and hyperbolic equations with arbitrary rough ($L^\infty$) coefficients. Our method does not rely on concepts of ergodicity or scale-separation but on compactness properties of the solution space and a new variational approach to homogenization. The approximation space is generated by an interpolation basis (over scattered points forming a mesh of resolution $H$) minimizing the $L^2$ norm of the source terms; its (pre-)computation involves minimizing $\mathcal{O}(H^{-d})$ quadratic (cell) problems on (super-)localized sub-domains of size $\mathcal{O}(H \ln (1/ H))$. The resulting localized linear systems remain sparse and banded. The resulting interpolation basis functions are biharmonic for $d\leq 3$, and polyharmonic for $d\geq 4$, for the operator $-\diiv(a
abla \cdot)$ and can be seen as a generalization of polyharmonic splines to differential operators with arbitrary rough coefficients. The accuracy of the method ($\mathcal{O}(H)$ in energy norm and independent from aspect ratios of the mesh formed by the scattered points) is established via the introduction of a new class of higher-order Poincare inequalities. The method bypasses (pre-)computations on the full domain and naturally generalizes to time dependent problems, it also provides a natural solution to the inverse problem of recovering the solution of a divergence form elliptic equation from a finite number of point measurements.
80 citations
TL;DR: In this article, the highest derivative appearing in the differential equation is expanded into the Haar series, this approximation is integrated while the boundary conditions are incorporated by using integration constants, and solutions to four test problems are investigated.
63 citations
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TL;DR: The WG method is introduced and analyzed in this article for solving the biharmonic equation in 2D and 3D and a refined Scott-Zhang interpolation operator is constructed to assist the corresponding error estimates.
Abstract: A C^0-weak Galerkin (WG) method is introduced and analyzed for solving the biharmonic equation in 2D and 3D. A weak Laplacian is defined for C^0 functions in the new weak formulation. This WG finite element formulation is symmetric, positive definite and parameter free. Optimal order error estimates are established in both a discrete H^2 norm and the L^2 norm, for the weak Galerkin finite element solution. Numerical results are presented to confirm the theory. As a technical tool, a refined Scott-Zhang interpolation operator is constructed to assist the corresponding error estimate. This refined interpolation preserves the volume mass of order (k+1-d) and the surface mass of order (k+2-d) for the P_{k+2} finite element functions in d-dimensional space.
62 citations
01 May 2012
TL;DR: In this article, the relationship between bi-harmonic maps and k-harmonics has been investigated, and a non-existence theorem of 3-harmonic maps has been shown.
Abstract: J. Eells and L. Lemaire introduced k-harmonic maps, and Wang Shaobo showed the first variational formula. When, k=2, it is called biharmonic maps (2-harmonic maps). There have been extensive studies in the area. In this paper, we consider the relationship between biharmonic maps and k-harmonic maps, and show non-existence theorem of 3-harmonic maps. We also give the definition of k-harmonic submanifolds of Euclidean spaces, and study k-harmonic curve in Euclidean spaces. Futhermore, we give a conjecture for k-harmonic submanifolds of Euclidean spaces.
53 citations
TL;DR: This work identifies characteristic artifacts caused by spurious local extrema, and provides a framework for minimizing quadratic energies on manifolds while constraining the solution to obey the maximum principle in the solved region.
Abstract: Functions that optimize Laplacian-based energies have become popular in geometry processing, eg for shape deformation, smoothing, multiscale kernel construction and interpolation Minimizers of Dirichlet energies, or solutions of Laplace equations, are harmonic functions that enjoy the maximum principle, ensuring no spurious local extrema in the interior of the solved domain occur However, these functions are only C0 at the constrained points, which often causes smoothness problems For this reason, many applications optimize higher-order Laplacian energies such as biharmonic or triharmonic Their minimizers exhibit increasing orders of continuity but lose the maximum principle and show oscillations In this work, we identify characteristic artifacts caused by spurious local extrema, and provide a framework for minimizing quadratic energies on manifolds while constraining the solution to obey the maximum principle in the solved region Our framework allows the user to specify locations and values of desired local maxima and minima, while preventing any other local extrema We demonstrate our method on the smoothness energies corresponding to popular polyharmonic functions and show its usefulness for fast handle-based shape deformation, controllable color diffusion, and topologically-constrained data smoothing © 2012 Wiley Periodicals, Inc
51 citations
TL;DR: In this article, the Bellows and Rulkov maps were investigated for vibrational resonance in two different nonlinear maps driven by a biharmonic force, and the results point at the relevance of the discrete-time models for the study of resonance phenomena, since analyses and simulations are much easier than for continuous time models.
50 citations
TL;DR: In this article, it was shown that a first order perturbation A ( x ) ⋅ D + q can be determined uniquely by measuring the Dirichlet-to-Neumann map on possibly very small subsets of the boundary of the domain.
49 citations
TL;DR: In this article, the authors give several construction methods and use them to produce many examples of proper bi-harmonic maps including biharmonic tori of any dimension in Euclidean spheres (Theorem 2.2, Corollaries 2.3, 2.4 and 2.6), bi-harmonic maps between spheres and into spheres via orthogonal multiplications and eigenmaps.
43 citations
TL;DR: In this paper, a new method to construct ''nice'' trial functions was introduced and a universal inequality for higher eigenvalues of the buckling problem was derived by making use of the trial functions.
Abstract: In this paper, we investigate an eigenvalue problem for a biharmonic operator on a bounded domain in an n-dimensional Euclidean space, which is also called a buckling problem. We introduce a new method to construct ``nice'' trial functions and we derive a universal inequality for higher eigenvalues of the buckling problem by making use of the trial functions. Thus, we give an affirmative answer for the problem on universal bounds for eigenvalues of the buckling problem, which was proposed by Payne, Polya and Weinberger in [14] and this problem has been mentioned again by Ashbaugh in [1].
TL;DR: In this article, the authors considered the numerical solution of initial boundary value problems for the harmonic (Laplacian) and biharmonic rotated diffusion operators, and provided stability criteria associated with the conventional space-time discretizations of the isoneutral Laplacians.
01 Dec 2012
TL;DR: Harmonic coordinates, which are solutions to the Laplace equation, provide a long list of desirable properties (making them suitable for a wide range of applications), but lack a closed‐form expression.
Abstract: Barycentric coordinates are an established mathematical tool in computer graphics and geometry processing, providing a convenient way of interpolating scalar or vector data from the boundary of a planar domain to its interior. Many different recipes for barycentric coordinates exist, some offering the convenience of a closed-form expression, some providing other desirable properties at the expense of longer computation times. For example, harmonic coordinates, which are solutions to the Laplace equation, provide a long list of desirable properties (making them suitable for a wide range of applications), but lack a closed-form expression.
We derive a new type of barycentric coordinates based on solutions to the biharmonic equation. These coordinates can be considered a natural generalization of harmonic coordinates, with the additional ability to interpolate boundary derivative data. We provide an efficient and accurate way to numerically compute the biharmonic coordinates and demonstrate their advantages over existing schemes. We show that biharmonic coordinates are especially appealing for (but not limited to) 2D shape and image deformation and have clear advantages over existing deformation methods. © 2012 Wiley Periodicals, Inc.
TL;DR: In this paper, a family of superlinear fourth order elliptic equations is proved to exist and to concentrate at a point in the limit, using variational methods and making use of a weak version of the Ambrosetti-Rabinowitz condition.
TL;DR: In this article, a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary is studied, and upper and lower bounds for the first eigen value in several contexts are derived.
TL;DR: In this paper, the problem of Gaussian beam scattering by finite arrays of pinned points, or platonic clusters, in a thin elastic plate governed by the biharmonic plate equation is considered.
Abstract: We consider the problem of Gaussian beam scattering by finite arrays of pinned points, or platonic clusters, in a thin elastic plate governed by the biharmonic plate equation. Integral representations for Gaussian incident beams are constructed and numerically evaluated to demonstrate the different behaviours exhibited by these finite arrays. We show that it is possible to extend the scattering theory from infinite arrays of pinned points to these finite crystals, which exhibit the predicted behaviour well. Analytical expressions for the photonic superprism parameters p, q and r, which are measures for dispersion inside the crystal, are also derived for the pinned plate problem here. We demonstrate the existence of negative refraction, beam splitting, Rayleigh anomalies, internal reflection, and near-trapping on the first band surface, giving examples for each of these behaviours.
TL;DR: For a semilinear biharmonic equation with exponential nonlinearity, the existence and the asymptotic behavior of entire solutions are studied in this article, and their stability and stability outside a compact set of Rn (n⩾2) is discussed in any space dimension n.
TL;DR: In this article, the elasticity solutions of an FGM rectangular plate with opposite edges simply supported under 12 types of biharmonic polynomial loads are derived as appropriate sums of the general and particular solutions of the governing equations.
TL;DR: V vibrating plates have many interesting applications, and from which the Chladni figures, representing sand ornaments which form on a vibrating plate, and the Tacoma Bridge are chosen, from which both the QR-algorithm and Lanczos can be well illustrated.
Abstract: Teaching linear algebra routines for computing eigenvalues of a matrix can be well moti- vated for students by using interesting examples. We propose in this paper to use vibrating plates for two reasons: First, they have many interesting applications, from which we chose the Chladni figures, representing sand ornaments which form on a vibrating plate, and the Tacoma Bridge, one of the most spectacular bridge failures. Second, the partial differential operator that arises from vibrating plates is the biharmonic operator, which one does not encounter often in a first course on numerical partial differential equations, and which is more challenging to discretize than the standard Laplacian seen in most textbooks. In addition, the history of vibrating plates is interesting, and we will show both spectral discretizations, leading to small dense matrix eigenvalue problems, and a finite difference discretization, leading to large scale sparse matrix eigenvalue problems. Hence both the QR-algorithm and Lanczos can be well illustrated.
TL;DR: In this article, the authors studied radially symmetric entire solutions of Δ 2 u = − u − q, u > 0 in R 3 and proved that for any q > 3 there exists a solution with exactly linear growth.
TL;DR: In this paper, the authors proved the nonexistence of a smooth stable solution to the biharmonic problem in the half space and the elliptic problem on a bounded smooth domain with the Navier boundary conditions.
Abstract: We prove the nonexistence of smooth stable solution to the biharmonic problem $\Delta^2 u= u^p$, $u>0$ in $\R^N$ for $1 5$ when $p > 1$, we obtain the nonexistence of smooth stable solution for any $N \leq 12$ and $p > 1$. Moreover, we consider also the corresponding problem in the half space $\R^N_+$, or the elliptic problem $\Delta^2 u= l(u+1)^p$ on a bounded smooth domain $O$ with the Navier boundary conditions. We will prove the regularity of the extremal solution in lower dimensions. Our results improve the previous works.
TL;DR: In this article, the existence of infinitely many weak solutions for a class of elliptic Navier boundary value problems depending on two parameters and involving the p -biharmonic operator was established.
Abstract: By using critical point theory, we establish the existence of infinitely many weak solutions for a class of elliptic Navier boundary value problems depending on two parameters and involving the p -biharmonic operator.
TL;DR: In this article, a partial classification result for proper biharmonic submanifolds with parallel mean curvature vector field in spheres is obtained, and then, completely classify the proper bi-harmonic subsets in spheres with parallel Mean Curvature Vector Field (MVF) and parallel Weingarten operator associated to the MGF.
TL;DR: Experimental results show that the proposed nonlinear elasticity model also needs fewer numerical corrections such as regridding steps for binary image registration, it renders better ground truth, and produces larger mutual information; most importantly, the landmark points distance and L2 dissimilarity measure between the gene expression data and corresponding mouse atlas are smaller compared with the registration model with biharmonic regularization.
Abstract: This paper proposes a numerical algorithm for image registration using energy minimization and nonlinear elasticity regularization. Application to the registration of gene expression data to a neuroanatomical mouse atlas in two dimensions is shown. We apply a nonlinear elasticity regularization to allow larger and smoother deformations, and further enforce optimality constraints on the landmark points distance for better feature matching. To overcome the difficulty of minimizing the nonlinear elasticity functional due to the nonlinearity in the derivatives of the displacement vector field, we introduce a matrix variable to approximate the Jacobian matrix and solve for the simplified Euler-Lagrange equations. By comparison with image registration using linear regularization, experimental results show that the proposed nonlinear elasticity model also needs fewer numerical corrections such as regridding steps for binary image registration, it renders better ground truth, and produces larger mutual information; most importantly, the landmark points distance and L 2 dissimilarity measure between the gene expression data and corresponding mouse atlas are smaller compared with the registration model with biharmonic regularization.
TL;DR: Finite volume schemes for the approximation of a biharmonic problem with Dirichlet boundary conditions are constructed and analyzed, first on grids which satisfy an orthogonality condition, and then on general, possibly non conforming meshes.
Abstract: Finite volume schemes for the approximation of a biharmonic problem with Dirichlet boundary conditions are constructed and analyzed, first on grids which satisfy an orthogonality condition, and then on general, possibly non conforming meshes. In both cases, the piece-wise constant approximate solution is shown to converge in L2 () to the exact solution; similar results are shown for the discrete approximate of the gradient and the discrete approximate of the Laplacian of the exact solution. Error estimates are also derived. These results are confirmed by numerical results.
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TL;DR: In this paper, the authors prove rigidity results for proper biharmonic immersions in hypersurfaces of the following types: Dupin hypersurface, both compact and non-compact, with bounded norm of the second fundamental form.
Abstract: We prove some new rigidity results for proper biharmonic immersions in ${\mathbb S}^n$ of the following types: Dupin hypersurfaces; hypersurfaces, both compact and non-compact, with bounded norm of the second fundamental form; hypersurfaces satisfying intrinsic properties; PMC submanifolds; parallel submanifolds.
TL;DR: In this paper, the authors consider the model problem for a plate described by a biharmonic equation and show that the target energy functional is differentiable with respect to a small perturbation of the crack length and the derivative can be represented as an invariant integral.
Abstract: We consider the model problem for a plate described by a biharmonic equation. The plate contains a rigid inclusion separated from the elastic part by a crack. We show that the target energy functional is differentiable with respect to a small perturbation of the crack length and the derivative can be represented as an invariant integral — a contour integral along a contour surrounding a crack vertex. The formula for the derivative and the invariant integral are analogues of the Griffith formula and Cherepanov–Rice integral known in the brittle fracture theory. Bibliography: 34 titles.
TL;DR: In this article, the authors studied stable solutions of a fourth order nonlinear elliptic equation, both in entire space and in bounded domains, and showed that stable solutions can be found in both spaces and domains.
Abstract: We study stable solutions of a fourth order nonlinear elliptic equation, both in entire space and in bounded domains.
TL;DR: By a careful inspection of the matrix elements of the discrete operator, it is shown that the boundary does not affect the approximation, and a full (“optimal”) fourth-order convergence is attained.
Abstract: It is well-known that non-periodic boundary conditions reduce considerably the overall accuracy of an approximating scheme. In previous papers the present authors have studied a fourth-order compact scheme for the one-dimensional biharmonic equation. It relies on Hermitian interpolation, using functional values and Hermitian derivatives on a three-point stencil. However, the fourth-order accuracy is reduced to a mere first-order near the boundary. In turn this leads to an "almost third-order" accuracy of the approximate solution. By a careful inspection of the matrix elements of the discrete operator, it is shown that the boundary does not affect the approximation, and a full ("optimal") fourth-order convergence is attained. A number of numerical examples corroborate this effect.
TL;DR: A new construction of a stable cubic spline-wavelet basis on the interval satisfying complementary boundary conditions of the second order is proposed, which means that the primal wavelet basis is adapted to homogeneous Dirichlet boundary conditions, while the dual wavelet based preserves the full degree of polynomial exactness.