Search

This paper considers the Dirichlet problem for the two-dimensional biharmonic equation in a bounded region consisting of a finite sum of rectangles. The biharmonic equation is first split into two Poisson equations and two classes of finite difference schemes are defined for obtaining the numerical solution. These classes correspond to the type of difference approximation defined for the missing boundary condition. Discretization error for the difference schemes in these two classes is shown to be of order $h^{{3 / 2}} $ and $h^2 $, respectively, as the mesh size $h \to 0$. The effect on discretization error of the different approximations within a class is also examined.

The Biharmonic Driver (BID) method uses direct (non-iterative) linear biharmonic solvers to obtain solutions to the nonlinear two-dimensional steady-state incompressible Navier-Stokes equations. Low Reynolds number recirculating flows are solved in 6–8 non-time-like iterations. Modifications are suggested for flow-through problems, and comparisons are made with other semidirect methods.

The quantum‐mechanical solution for the biharmonic oscillator is presented. This one‐dimensional asymmetric oscillator is described by the potential energy function V(x) = mω12x2/2, x ≳ 0, and V(x) = mω22x2/2, x

SummaryA double series solution of the biharmonic equation is derived in the rectangular domain. The solution utilizes the finite double Fourier sine transform and the analysis shows how the transform may be evaluated for quite arbitrary boundary conditions, the solution then being found from the corresponding inversion formula.The method is applied to some classical boundary value problems which have been investigated by other means by earlier workers. It is felt that some of the series solutions given here are more amenable to computation than some of the earlier methods.In some of the problems considered, notably the elastostatics of the rectangular plate and the bending of clamped plates, the method reduces to solving an infinite set of linear equations. In such cases, a finite leading matrix of the coefficients has to be considered. In the examples considered, this technique produced results which converged to known solutions obtained by other methods. However, the mathematical criteria which permit the infinite system to be curtailed, have not been established.

A finite element approximation of the Dirichlet problem for the biharmonic operator is described. Its main feature is that it is equivalent to solving a sequence of discrete Dirichlet problems for the operator -Δ. This method, which has already been shown to be convergent, is particularly well-suited for problems in fluid dynamics.

In this paper we compute the motion and the shape of the free surface on a liquid in a trench heated from its side. The analysis is based on Joseph's Lagrangian theory of domain perturbations, which is developed in general and through simple examples, chosen so as to make the comparison of the Lagrangian method with Stokes's Eulerian theory very clear. The perturbation problems are resolved analytically by application of biorthogonality conditions to a powerful set of biharmonic eigenfunctions.

The biharmonic boundary value problem with Dirichlet boundary conditions is reduced to a coupled system of Poisson equations, which depends upon an arbitrary, positive coupling constant c. Since each of the Poisson equations is well-posed, the system may be solved by iteration. We show that the iterates may be represented as a linear combination of the eigenfunctions of the “Dirichlet eigenvalue problem” (a fourth order boundary value problem with the eigenvalue in the boundary condition). Convergence of the iterative scheme occurs when $0 < c < 2v_1$, where $v_1 $ is the smallest eigenvalue. By making use of an averaging scheme, convergence may be produced for any positive c. With the proper choice of c, the rate of convergence may be increased. This coupled equation approach includes the finite difference approach as a special case.

Nonsymmetric linear systems are by far not as common as syemmtric ones but nevertheless systems with nonsymmetric matrices appear, e. g., in the numerical solution of the biharmonic equation, the computation of splines or the solution of some special integral equations. The SOR-method applied to linear systems X = BX + C with skew-symmetric matrix B is studied. Described is a region in the complex plane which contains the eigenvalues of the SOR-operator. Using this information a relaxed SOR-method is proposed; bounds for the spectral radius of the iteration operator are derived. The advantage is that the values of the corresponding iteration parameters can be directly calculated from the norm of the given matrix.

The Dirichlet problem for biharmonic equation in a rectangular region is considered. The method of splitting is used and two classes of finite difference approximations are defined. Two semi-iterative procedures are considered for obtaining the solution of the resulting coupled system of algebraic equations. It is shown that the rate of convergence of the iterative procedures depends upon the choice of the difference approximation. Estimates for optimum iteration parameters are given and several comparisons are made. An attempt is made to unify the ideas on the splitting technique for solving the first biharmonic boundary value problem.

The distortions of a thrust pad and its runner are analysed by solving the biharmonic equation of plate flexure by both elastic and thermal causes. The temperature and pressures are obtained from the simultaneous solution of the Reynolds and energy equations. The major similification is the assumption of constant temperature through the thickness of the oil film, though it varies in both the other directions. The equations are set up and their reduction to a form suitable for computing is described. Four modes of distortion are considered, pressure and thermal bending, direct compression and direct thermal expansion.

Two-dimensional viscous flow due to the motion of a semi-infinite flat plate moving perpendicular to its own plane in a region of fluid bounded by a rigid plane well is considered by means of the Stokes approximation. The stream function, being biharmonic, can be constructed by use of two harmonic functions. By mapping the flow region conformally onto the upper half of a complex plane, the problem can be reduced to determining the harmonic functions in this plane. From the first and second approximate solutions are obtained a lower and an upper bound, respectively, of the discharge of fluid through the gap between the plate and the well. The case in which the semi-infinite plate is held at rest and the flow is caused by the pressure difference through the gap is also considered.

A complete characterization is given for the fundamental solution of the complex-valued differential operator governing the linear behavior of a shallow, elastically isotropic hyperbolic paraboloidal shell. The only previously known fundamental solutions in shell theory have been those for the sphere and the shallow cylinder. Physically, the real and imaginary parts of these fundamental solutions represent the normal deflection and Airy’s stress function for an unbounded shell under a normal point load. The differential operator for the hyperbolic paraboloid consists of the biharmonic operator minus the imaginary unit i times the second order wave operator. Unlike the case with the sphere and the cylinder, this operator cannot be factored, and this accounts for the relative complexity of the analysis. The fundamental solution is expressed as a Fourier series in the polar angle, and the radially dependent Fourier coefficients are represented by a contour integral. From this, explicit series and asymptotic ...

The discrete biharmonic equation on a rectangular region and the discrete Poisson equation on an irregular region can be treated as modifications to matrix problems with very special structure. We show how to use the direct method of matrix decomposition to formulate an effective numerical algorithm for these problems. For typical applications the operation count is $O(N^3 )$ for an $N \times N$ grid. Numerical comparisons with other techniques are included.

Several forms of “Successive Overrelaxation (SOR)” are applied to a coupled pair of difference equations, equivalent to the biharmonic equation. The analysis and results yield a partial comparison amongst the methods. Computational results are also included.

This paper presents a method of solution for an infinite wedge containing a symmetrically located circular hole. The solution is formulated separately according to the given in-plane edge tractions being even or odd with respect to the axis of the wedge. In either case, the stress function is constructed as the sum of four parts of biharmonic functions, two in the form of integrals and the other two in the form of series, in addition to a basic stress function for an otherwise unperforated wedge. The four parts as a whole give no traction along the edges and no stress at infinity of the wedge. Together with the basic stress function, the boundary conditions of no traction at the rim of hole are adjusted. Complex expressions are used in adjusting the boundary conditions. Finally, numerical examples are given for illustration.

The mathematical theory of thin elastic plates loaded by transverse forces leads to biharmonic boundary value problems. These may be formulated in terms of singular integral equations, which can be solved numerically to a tolerable accuracy for any shape of boundary by digital computer programs. Particular attention is devoted to clamped and simply-supported rectangular plates. Our results indicate support for the generally accepted treatment of such plates and for the intuitive picture of deflection behaviour at a corner.

A numerical procedure is proposed for the solution of the biharmonic equation for deflections of uniformly loaded plates with all edges clamped. It is shown that the quantity ∇2w at the plate boundary can be determined numerically by noting a linear relationship between ∇2w and w in finite-difference form at the boundary. According to the classical methods of solution, the quantity ∇2w at the clamped edge cannot be determined in advance. The new approach is based on the boundary vorticity method for the vorticity transport equation governing a steady two-dimensional incompressible fluid flow. By using the new procedure, it is now possible to replace the biharmonic equation for clamped plates by two second-order partial differential equations of elliptic type similar to those for simply supported plates with straight edges. The possibility of extending the procedure to various problems in plates and shells is pointed out. The study also serves to illustrate a close relationship between a class of fluid mechanics problems and a class of plate and shell problems.

The two-dimensional biharmonic operator is approximated by a finite-difference operator over a square (h x A) net, which connects each node with 16 neighbouring nodes in such a manner that the resulting matrix has "Young's Property A," for simple boundary conditions. It is shown that the local truncation error, the convergence rate of S.O.R. for solving the finite-difference (or "net") equations, and the truncation error of the solution of the net equations are each of order O(A) as h -> 0. Comparisons are made with the conventional 13-node approximation to the biharmonic operator: in particular, numerical experiments indicate that the convergence rate of S.O.R. for solving net equations based on the conventional 13-node operator is considerably less than that for the novel 17-node operator.

In this paper, we will show that the strict monotonicity of the eigenvalues of the biharmonic operator holds if and only if some unique continuation property is satisfied by the corresponding eigenfunctions.

A formulation of two-dimensional magneto-thermo-elastostatic boundary value problems is presented in this paper. It is found that since the Lorentz pondermotive body force is conservative in this case, a nonhomogeneous biharmonic equation for the stress function can be derived. The effects of the Lorentz body force and Joule heating on the elastic deformation can thus be easily examined. As an illustrative example, stresses induced in an infinitely long cylindrical conductor carrying a uniformly distributed axial current is analysed.

It is shown that biharmonic boundary value problems can be formulated in terms of integral equations. Procedures are given for solving such equations numerically, for smoothing out sharp corners on the boundary, and for checking results. These techniques are applied to attack some two-dimensional stress problems which do not seem to be amenable to any other treatment.

Stress analysis for a circular cylindrical shell having two circular cutouts of equal size is worked out. The following four cases are treated. A circular cylindrical shell having two circular cutouts of equal size (a) in a longitudinal direction under uniform tension, (b) in a circumferential direction under uniform tension, (c) in a longitudinal direction under internal pressure and (d) in a circumferential direction under internal pressure. The method of Green's general biharmonic analysis is extended to this problem and the boundary conditions along the edge of cutouts are satisfied by the point-matching method. Numerical solutions are obtained for three kinds of diameter of circular cutout and four kinds of center-distance of the cutouts. Especially, in the cases of (a) and (b), the results are compared with those of the flat plate.

Die Kelvin-Almansi-Darstellung einer Bipotentialfunktion vermittelst zweier harmonischer Funktionen wird zur Losung elastischer Warmespannungsprobleme fur stationare Temperaturfelder angewendet.

An analogue technique is described giving good approximate solutions of the biharmonic equation. The method uses two plane sheets of conducting paper linked by an array of resistors. The technique is applied to problems involving the slow flow of a viscous fluid past uniformly spaced parallel fibres. Calculated values of the Kozeny constant from the results show much better agreement with the practical and theoretical values than those derived from previous analogue solutions.

It is proved that a function which is biharmonic in a half-strip and satisfies homogeneous Dirichlet conditions at the base of the strip oscillates.

A solution is presented for the computation of the transient thermoelastic stresses in a hollow cylinder with temperature boundary conditions given as a circumferential variation of surface heat transfer coefficient. The temperature distribution is solved explicitly. The problem is set up using the Airy stress function which leads to the biharmonic equation. This approach requires the satisfaction of three Michell integrals at the inner boundary in order to ensure single-valued displacements and rotation. An iterative method is described in which these integrals are all simultaneously satisfied and thus provide the necessary non-zero boundary conditions for the solution of the biharmonic equation which is rapidly solved by Gaussian elimination. Results are presented for the general case where the temperature is a function of r and θ. The computer program is checked by assuming a constant value of the surface heat transfer coefficients. In this case a closed form solution is obtained.

The Kirchhoff assumption in thin elastic plates results in a biharmonic equation for the lateral deflection and a C 1 deflection field is therefore required in the finite element method for their approximate solution. By considering the thin plate as a three dimensional solid and by discarding the Kirchhoff assumption, the continuity requirement for the displacements is reduced to C 0 . The stiffness matrix produced in this way becomes, however, violently ill-conditioned as the thickness t of the structure is reduced. It is shown here that the factor 1/t 2 causing this ill-conditioning can be removed from the stiffness matrix and consequently from its condition number by relating the thickness t to the diameter of the element h, without losing the rate of convergence provided by the degree of the shape functions inside the element. This is used here to construct a well-conditioned 9-degrees-of-freedom plate bending element which is only C 0 but which converges quadratically to the C 1 solution (Kirchhoff solution) of thin plates. Addition of shear to C 1 elements is also considered.