The use of the biharmonic operator for deforming a mesh in an arbitrary-Lagrangian-Eulerian simulation is investigated. The biharmonic operator has the advantage that two conditions can be specified on each boundary of the mesh. This allows both the position and the normal mesh spacing along a boundary to be controlled, which is important for two-fluid interfaces and periodic boundaries. At the...

In this paper, we formulate a novel trivariate biharmonic B-spline defined over bounded volumetric domain. The properties of bi-Laplacian have been well investigated, but the straightforward generalization from bivariate case to trivariate one gives rise to unsatisfactory discretization, due to the dramatically uneven distribution of neighbouring knots in 3D. To ameliorate, our original idea is...

Biharmonic equations have many applications, especially in fluid and solid mechanics, but difficult to solve due to the fourth order derivatives in the differential equation. In this paper a fast second order accurate algorithm based on a finite difference discretization and a Cartesian grid is developed for two dimensional biharmonic equations on irregular domains with essential boundary condi...

Biharmonic equations have many applications, especially in fluid and solid mechanics, but difficult to solve due to the fourth order derivatives in the differential equation. In this paper a fast second order accurate algorithm based on a finite difference discretization and a Cartesian grid is developed for two dimensional biharmonic equations on irregular domains with essential boundary condi...

We study the behavior of two maps in an effort to better understand the stability of ωlimit sets ω(x, f ) as we perturb either x or f , or both. The first map is the set-valued function Λ taking f in C(I ,I) to its collection of ω-limit points Λ( f ) = ⋃x∈I ω(x, f ), and the second is the map Ω taking f in C(I ,I) to its collection of ω-limit sets Ω( f )= {ω(x, f ) : x ∈ I}.We characterize thos...

In this paper, we first establish a Landau-Bloch type theorem for certain bounded and normalized biharmonic mappings F(z) = |z|21(z) + h(z), where 1(z) h(z) are harmonic in the unit disk with |1(z)| ? M1, |h(z)| M2. particular, our result is sharp when M1 M2 1. Then, several new versions of theorems coefficients condition place or obtain results.

Let denote the unit circle in the complex plane. Given a function , one uses t usual (harmonic) Poisson kernel for the unit disk to define the Poisson integral of , namely . Here we consider the biharmonic Poisson kernel for the unit disk to define the notion of -integral of a given function ; this associated biharmonic function will be denoted by . We then consider the dilations ...

B.Y. Chen introduced biharmonic submanifolds in Euclidean spaces and raised the conjecture ”Any biharmonic submanifold is minimal”. In this article, we show some affirmative partial answers of generalized Chen’s conjecture. Especially, we show that the triharmonic hypersurfaces with constant mean curvature are minimal. M.S.C. 2010: 58E20, 53C43.