Fourth-order Elliptic Partial Differential Equations Research Articles

The Kirchhoff plate equation on surfaces: the surface Hellan–Herrmann–Johnson method

Abstract We present a mixed finite element method for approximating a fourth-order elliptic partial differential equation (PDE), the Kirchhoff plate equation, on a surface embedded in ${\mathbb {R}}^{3}$, with or without boundary. Error estimates are given in mesh-dependent norms that account for the surface approximation and the approximation of the surface PDE. The method is built on the classic Hellan–Herrmann–Johnson method (for flat domains), and convergence is established for $C^{k+1}$ surfaces, with degree $k$ (Lagrangian, parametrically curved) approximation of the surface, for any $k \geqslant 1$. Mixed boundary conditions are allowed, including clamped, simply-supported and free conditions; if free conditions are present then the surface must be at least $C^{2,1}$. The framework uses tools from differential geometry and is directly related to the seminal work of Dziuk, G. (1988) Finite elements for the Beltrami operator on arbitrary surfaces. Partial Differential Equations and Calculus of Variations, vol. 1357 (S. Hildebrandt & R. Leis eds). Berlin, Heidelberg: Springer, pp. 142–155. for approximating the Laplace–Beltrami equation. The analysis here is the first to handle the full surface Hessian operator directly. Numerical examples are given on nontrivial surfaces that demonstrate our convergence estimates. In addition, we show how the surface biharmonic equation can be solved with this method.

Facial geometry parameterisation based on Partial Differential Equations

Geometric modelling using Partial Differential Equations (PDEs) has been gradually recognised due to its smooth instinct, as well as the ability to generate a variety of geometric shapes by intuitively manipulating a relatively small set of PDE boundary curves. In this paper we explore and demonstrate the feasibility of the PDE method in facial geometry parameterisation. The geometry of a generic face is approximated by evaluating spectral solutions to a group of fourth order elliptic PDEs. Our PDE-based parameterisation scheme can produce and animate a high-resolution 3D face with a relatively small number of parameters. By taking advantage of parametric representation, the PDE method can use one fixed animation scheme to manipulate the facial geometry in varying Levels of Detail (LODs), without any further process.

Numerical performance of parallel group explicit solvers for the solution of fourth order elliptic equations

Many applications in applied mathematics and engineering involve numerical solutions of partial differential equations (PDEs). Various discretisation procedures such as the finite difference method result in a problem of solving large, sparse systems of linear equations. In this paper, a group iterative numerical scheme based on the rotated (skewed) five-point finite difference discretisation is proposed for the solution of a fourth order elliptic PDE which represents physical situations in fluid mechanics and elasticity. The rotated approximation formulas lead to schemes with lower computational complexities compared to the centred approximation formulas since the iterative procedure need only involve nodes on half of the total grid points in the solution domain. We describe the development of the parallel group iterative scheme on a cluster of distributed memory parallel computer using Message-Passing Interface (MPI) programming environment. A comparative study with another group iterative scheme derived from the centred difference formula is also presented. A detailed performance analysis of the parallel implementations of both group methods will be reported and discussed.

A parallel mixed finite element implementation of fourth-order plate bending problems in distributed memory environments

The paper deals with a parallel implementation of a mixed finite element method of solution of general fourth-order elliptic partial differential equations which in the case of aniso-/ortho-/isotropic plate bending problems simultaneously approximates the bending moment tensor field Ψ=( ψ ij ) 1⩽ i, j⩽2 and the displacement field u. The experiments on the performance of this method have been done in two parallel environments namely, Silicon Graphics Origin 3800 and Cray T3E. The details of the parallel computer implementation procedures for this mixed method with time, speed-up comparisons for a few illustrative examples are given.

A parallel mixed finite element implementation of fourth-order plate bending problems in distributed memory environments

The paper deals with a parallel implementation of a mixed finite element method of solution of general fourth-order elliptic partial differential equations which in the case of aniso-/ortho-/isotropic plate bending problems simultaneously approximates the bending moment tensor field Ψ =( ψ ij ) 1⩽ i , j ⩽2 and the displacement field u . The experiments on the performance of this method have been done in two parallel environments namely, Silicon Graphics Origin 3800 and Cray T3E. The details of the parallel computer implementation procedures for this mixed method with time, speed-up comparisons for a few illustrative examples are given.

A Parameter Identification Problem of Mixed Type Related to the Manufacture of Car Windshields

We study the identification of a parameter in a fourth-order elliptic partial differential equation that models the optimal design of car windshields to be manufactured by the sagging process. Considered as a second-order equation for the unknown parameter, the problem is of mixed type, i.e., changing between elliptic and hyperbolic. Numerical routines for directly solving this equation are not available. In this paper we both theoretically and numerically show that the inverse problem can instead be solved in a stable way by means of a (derivative free) iterative regularization method. The course of the iteration nevertheless depends markedly on the mixed type of the second-order equation.

Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity

A new finite element method for fourth-order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem. The method combines concepts from the continuous Galerkin (CG) method, the discontinuous Galerkin (DG) method and stabilization techniques. A brief review of the CG method, the DG method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth-order elliptic problems. A continuous/discontinuous Galerkin (C/DG) method is proposed which uses C 0-continuous interpolation functions and is formulated in the primary variable only. The advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided. In the context of thin bending theory, the C/DG method leads to a formulation where displacements are the only degrees of freedom, and no rotational degrees of freedom need to be considered. The main feature of the C/DG method is the weak enforcement of continuity of first and higher-order derivatives through stabilizing terms on interior boundaries. Consistency, stability and convergence of the method are shown analytically. Numerical experiments verify the theoretical results, and applications are presented for Bernoulli–Euler beam bending, Poisson–Kirchhoff plate bending and a shear layer problem using Toupin–Mindlin strain gradient theory.

An Efficient ADI-Solver for Scattered Data Problems with Global Smoothing

For the approximate representation of large data sets over a parameter domain inR2, many geological and other applications require the construction of surfaces which have minimal energy, i.e., minimal curvature. One way to achieve this is by solving a fourth-order elliptic partial differential equation. Its discretization by a difference scheme makes it particularly easy to incorporate (appropriate approximations of) known data points. In this paper, we investigate the performance of different solution methods for the resulting symmetric linear system of equations since this is the most CPU-demanding step in the scattered data approximation procedure. Specifically, we test first the performance of a preconditioned conjugate gradient method with an SSOR and an RILU preconditioner. However, since the partial differential operator does not contain mixed derivatives, using an alternating-direction-implicit scheme (ADI method) which has been employed successfully in the past for second-order problems, together with a Cholesky factorization of the corresponding one-dimensional operators has also been tried for the problem at hand. The computational studies that we have performed here show that for our problem an instationary ADI method is superior to the above-mentioned preconditioned conjugate gradient solvers both with respect to work load and accuracy of the solution. For the fourth-order model problem considered in this paper, the instationary ADI method with Wachspress parameters results in a number of iterations that is essentially independent of the number of variables.

Estimation of the conditional stability of a two-contour boundary-value problem

We obtain estimates for the conditional stability of a two-contour boundary-value problem for fourth-order elliptic partial differential equations.

SAINT-VENANTEND EFFECTS FOR PLANE DEFORMATIONS OF ELASTIC COMPOSITES

SUMMARY In the linear theory of elasticity, Saint-Venant's principle is used to justify the neglect of edge effects when determining stresses in a body. For isotropic materials, the validity of this is well established. However, for anisotropic and composite materials, experimental results have shown that edge effects may persist much farther into the material than for isotropic materials and as a result cannot be neglected. This paper provides a further analysis of the effects of material anisotropy on the exponential decay rate for stresses in a semi-infinite elastic strip. A linearly elastic semi-infinite strip in a state of plane stress/strain subject to a self-equilibrated end load is considered first for a specially orthotropic material and then for the general anisotropic material. The problem is governed by a fourth-order elliptic partial differential equation with constant coefficients. In the former case just a single dimension-less material parameter appears, while in the latter, only three dim...

End effects for plane deformations of an elastic anisotropic semi-infinite strip

In the linear theory of elasticity, Saint-Venant's principle is used to justify the neglect of edge effects when determining stresses in a body. For isotropic materials, the validity of this is well established. However for anisotropic and composite materials, experimental results have shown that edge effects may persist much farther into the material than for isotropic materials and as a result cannot be neglected. This paper further examines the effects of material anisotropy on the exponential decay rate for stresses in a semi-infinite elastic strip. A linearly elastic semi-infinite strip in a state of plane stress/strain subject to a self-equilibrated end load is considered first for a specially orthotropic material and then for the general anisotropic material. The problem is governed by a fourth-order elliptic partial differential equation with constant coefficients. In the former case, just a single dimensionless material parameter appears, while in the latter, only three dimensionless parameters are required. Energy methods are used to establish lower bounds on the actual stress decay rate. Both analytic and numerical estimates are obtained in terms of the elastic constants of the material and results are shown for several contemporary engineering materials. When compared with the exact stress decay rate computed numerically from the eigenvalues of a fourth-order ordinary differential equation, the results in some cases show a high degree of accuracy. In particular, for strongly orthotropic materials, an asymptotic estimate provides extremely accurate estimates for the decay rate. Results of the type obtained here have several important practical applications. For example, they provide physical insight into the mechanical testing of anisotropic and laminated composite structures (including the off-axis tension test), are useful in assessing the influence of fasteners, joints, etc. on the behavior of composite structures and allow for “tailoring” a material with specific properties to ensure that local stresses attenuate at a desired rate.

A mixed finite-element method for fourth-order elliptic problems with variable coefficients

A new mixed finite-element method for the solution of the Dirichlet problem of fourth-order elliptic partial differential equations with variable coefficients on a convex polygon has been developed in this paper. Biharmonic and bending problems of elastic plates, for which this technique allows a simultaneous approximation to the deflection, components of the change in curvature tensor and the bending and twisting moments, are the particular cases of the problem considered in the paper. Error estimate for the mixed finite-element solution has been given.

Asymptotic Solutions of the First Boundary Value Problem for a Fourth-Order Elliptic Partial Differential Equation

Asymptotic Solutions of the First Boundary Value Problem for a Fourth-Order Elliptic Partial Differential Equation