A mixed formulation with two main variables, based on the Ciarlet-Raviart technique, with 0 C continuity shape functions is employed for the solution of some types of biharmonic equations in 1-D. The continuous and discrete Babuška-Brezzi inf-sup conditions are established. The formulation is numerically tested for both the hand pextensions. The model problems involve the standard biharmonic eq...

Recently the biharmonic form of the Navier-Stokes (N-S) equations have been solved in various domains by using second order compact discretization. In this paper, we present a fourth order essentially compact (4OEC) finite difference scheme for the steady N-S equations in geometries beyond rectangular. As a further advancement to the earlier formulations on the classical biharmonic equation tha...

In this work, we compare different mesh moving techniques for monolithically-coupled fluid-structure interactions in arbitrary Lagrangian–Eulerian coordinates. The mesh movement is realized by solving an additional partial differential equation of harmonic, linear-elastic, or biharmonic type. We examine an implementation of time discretization that is designed with finite differences. Spatial d...

In this paper we present a method for generating Bézier surfaces from the boundary information based on a general 4th-order PDE. This is a generalisation of our previous work on harmonic and biharmonic Bézier surfaces whereby we studied the Bézier solutions for Laplace and the standard biharmonic equation, respectively. Here we study the Bézier solutions of the Euler–Lagrange equation associate...

We develop a simple and efficient FFT-based fast direct solver for the biharmonic equation on a disk. The biharmonic equation is split into a coupled system of harmonic problems. We first use the truncated Fourier series expansion to derive a set of coupled singular ODEs, then we solve those singular equations by second-order finite difference discretizations. Using a radial grid with shifting ...

Abstract. Using Maz’ya type integral identities with power weights, we obtain new boundary estimates for biharmonic functions on Lipschitz and convex domains in R. For n ≥ 8, combined with a result in [S2], these estimates lead to the solvability of the L Dirichlet problem for the biharmonic equation on Lipschitz domains for a new range of p. In the case of convex domains, the estimates allow u...

In this paper, the well-known nonconforming Morley element for biharmonic equations in two spatial dimensions is extended to any higher dimensions in a canonical fashion. The general -dimensional Morley element consists of all quadratic polynomials defined on each -simplex with degrees of freedom given by the integral average of the normal derivative on each -subsimplex and the integral average...

This article introduces and analyzes a weak Galerkin mixed finite element method for solving the biharmonic equation. The weak Galerkin method, first introduced by two of the authors (J. Wang and X. Ye) in [52] for second order elliptic problems, is based on the concept of discrete weak gradients. The method uses completely discrete finite element functions and, using certain discrete spaces an...

The computation of curvature quantities over discrete geometry is often required when processing geometry composed of meshes. Curvature information is often important for the purpose of shape analysis, feature recognition and geometry segmentation. In this paper we present a method for accurate estimation of curvature on discrete geometry especially those composed of meshes. We utilise a method...

The coefficients for a nine-point high-order accuracy discretization scheme for a biharmonic equation∇4u = f(x, y) (∇2 is the two-dimensional Laplacian operator) are derived. The biharmonic problem is defined on a rectangular domain with two types of boundary conditions: (1) u and ∂u/∂n or (2) u and ∂u/∂n (where ∂/∂n is the normal to the boundary derivative) are specified at the boundary. For b...