Biharmonic equation

The system of equations with initial and boundary conditions formulated above allows us to find the velocity distributions and pressure drop for the filled part of the mold. In order to incorporate effects related to the movement of the stream front and the fountain effect, it is possible to use the velocity distribution obtained285 for isothermal flow of a Newtonian liquid in a semi-infinite plane channel, when the flow is initiated by a piston moving along the channel with velocity uo (it is evident that uo equals the average velocity of the liquid in the channel). An approximate quasi-stationary solution can be found. Introduction of the function v /, transforms the momentum balance equation into a biharmonic equation. Then, after some approximations, the following solution for the function jt was obtained 285... 

The stresses around a crack tip are most easily computed from the usual stress function, 0, which must satisfy the biharmonic equation ... 

Airy Stress Eunction and the Biharmonic Equation The biharmonic equation in many instances has an analogous role in continuum mechanics to that of Laplace s equation in electrostatics. In the context of two-dimensional continuum mechanics, the biharmonic equation arises after introduction of a scalar potential known as the Airy stress function f such that... 

Compatibility conditions address the fact that the various components of the strain tensor may not be stated independently since they are all related through the fact that they are derived as gradients of the displacement fields. Using the statement of compatibility given above, the constitutive equations for an isotropic linear elastic solid and the definition of the Airy stress function show that the compatibility condition given above, when written in terms of the Airy stress function, results in the biharmonic equation = 0. 

We saw, in the previous section, that problems of creeping-motion in two dimensions can be reduced to the solution of the biharmonic equation, (7 46), subject to appropriate boundary conditions. To actually obtain a solution, it is convenient to express (7 46) as a coupled pair of second-order PDEs ... 

In other words, the real and imaginary parts of analytic functions are harmonic and would satisfy the biharmonic equation (see Eqn. (3.14)). The task then becomes one of identifying the appropriate analytic functions that can satisfy the boundary conditions of the problem. The methodology is applied to the solution of crack problems. 

Because the derivatives of analytic functions are also analytic and are harmonic, the chosen function th(z) satisfies the biharmonic equation = 0. Note also that Eqn. (3.21) is a special case of Eqn. (3.15), in which only the first and third functions are retained, namely ... 

In such a two dimensional geometry, the Stokes equations are reduced to the biharmonic equation for the stream function... 

Thus, any solution to Eq. (4.24) that also fits the boundary conditions will be the elastic solution to the problem being sought. Conversely, mathematical functions that satisfy Eq. (4.23) are often studied to find the associated elastic problem. The function x is called the Airy stress function and Eq. (4.24) is called the Biharmonic equation This latter equation does not require any elastic constants for its solution, indicating the stress distributions are independent of the elastic properties. If, however, the strains are needed, the elastic constants appear once Hooke s Law is introduced. 

Hence, in order to determine the deviation from spherical shape, it was necessary to include higher order terms proportional to IP in the stream function. However, such a term emphasizes inertial forces which are not accounted for in the biharmonic equation (Equation 7.21). Proudman and Pearson (1957) have shown how to correct the biharmonic equation for small effects of inertia and obtained the solution for a perfect sphere. The stream function takes the formt Ji + Nr,. ... 

An Adaptive Algorithm for Solving the Biharmonic Equation on Sparse Grids... 

This paper presents a sparse grid algorithm with higher order elements for solving generalized problems of the biharmonic equation. Furthermore, an adaptive version of the algorithm together with numerical results will be discussed. 

Selvadurai APS (2000a) Partial differential equations in mechanics, vol 1. Fundamentals, Laplace s equation, diffiision equation, wave equation. Springer, Berhn Selvadurai APS (2(XX)b) Partial differential equations in mechanics, vol 2. The biharmonic equation, Poisson s equation. Springer, Berhn... 

A Boundary Element Method (B.E.M.) technique has been applied to effect a solution to the problem of low Reynolds number flow at the inlet to a thrust pad bearing. By the introduction of a suitable stream function, the Biharmonic Equation, = 0, was obtained and solved in the inlet region. 

The introduction of the stream function is such that the continuity equation (2) is automatically satisfied. Equation (1) now reduces to the Biharmonic Equation,... 

Altas I, Dym J, Gupta MM, Manohar RP (1998) Multigiid solution of automatically generated high order discretizations for the biharmonic equation. SIAM J Sci Comput 19 1575—1585... 

Another popular and useful approach for many practical engineering problems that can be reduced to two dimensional plane strain or plane stress approximations involves an auxiliary stress potential. In this approach, a bi-harmonic equation is developed based on the stresses (in terms of the potential) satisfying both the equilibrium equation and the compatibility equations. The result is that stresses derived from potentials satisfying the biharmonic equation automatically satisfy the necessary field equations and only the boundary conditions must be verified for any given problem. A rich set of problems may be solved in this manner and examples can be found in many classical texts on elasticity. In conjunction with the use of the stress potential, the principle of superposition is also often invoked to combine the solutions of several relatively simple problems to solve quite complex problems. 

All the elasticity equations given by Eqs. 9.29 - 9.32 as well as the biharmonic stress function equation can be developed for cylindrical coordinates (see, Timoshenko and Goodier, (1970)). The biharmonic equation is written as,... 

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