Implementation of MB algorithm for nonlinear magnetostatics

The iteration strategy in the MB formulation for nonlinear magnetostatic problems is discussed. The difference between MH and MB iteration is examined, and two methods for implementating the constitutive M(B) relation are presented.     

Some improvements in nonlinear 3D magnetostatics

It is pointed out that nonlinear magnetostatic field problems must be solved by an iterative approach, either by direct iteration or by the Newton-Raphson method or by a combination of both. Some improvements are described here to decrease the number of iterations that are necessary to arrive at reliable results. CPU-time requirements with and without these improvements are compared. The improved software package is applied to an inverse problem, in which the form of the pole of a C-shaped nonlinear magnet is to be optimized     

Implicit formulation with the boundary element method for nonlinear radiation of water waves

An accurate and efficient numerical method is presented for the two-dimensional nonlinear radiation problem of water waves. The wave motion that occurs on water due to an oscillating body is described under the assumption of ideal fluid flow. The governing Laplace equation is effectively solved by utilizing the GMRES (Generalized Minimal RESidual) algorithm for the boundary element method (BEM) with quadratic approximation. The intersection or corner singularity in the mixed Dirichlet–Neumann problem is resolved by introducing discontinuous elements. The fully implicit trapezoidal rule is used to update solutions at new time-steps, by considering stability and accuracy. Traveling waves generated by the oscillating body are absorbed downstream by the damping zone technique. To avoid the numerical instability caused by the local gathering of grid points, the re-gridding technique is employed, so that all the grids on the free surface may be re-distributed with an equal distance between them. The nonlinear radiation force is evaluated by means of the acceleration potential. For a mixed Dirichlet–Neumann problem in a computational domain with a wavy top boundary, the present BEM yields numerical solutions for the quadratic rate of convergence with respect to the number of boundary elements. It is also demonstrated that the present time-marching and radiation condition work successfully for nonlinear radiation problems of water waves. The results obtained from this study concur reasonably well with other numerical computations.     

Notes on boundary integral equations for three-dimensional magnetostatics

Several methods of formulating two-region magnetostatics problems with boundary integral equations and scalar potentials are discussed. These integral equations contain single- (i.e., monopole) and/or double- (i.e., dipole) layer source distributions. If only one type of source is used for both regions, certain numerical difficulties occur, which are discussed. For numerical accuracy a combined approach is adopted: it is sufficient to choose single layers in the exterior or less permeable region and to use double layers in the interior high-permeability region and at the interface. As an example, the combined method is applied to a recording head energized with a current loop.     

Numerical solution of a parabolic system with coupled nonlinear boundary conditions

A numerical technique has been developed to solve a system that consists of m linear parabolic differential equations with coupled nonlinear boundary conditions. Such a system may represent chemical reactions, chemical lasers and diffusion problems. An implicit finite difference scheme is adopted to discretize the problem, and the resulting system of equations is solved by a novel technique that is a modification of the cyclic odd–even reduction and factorization (CORF) algorithm. At each time level, the system of equations is first reduced to m nonlinear algebraic equations that involve only the m unknown grid points on the nonlinear boundary. Newton's method is used to determine these m unknowns, and the corresponding Jacobian matrix can be computed and updated easily. After convergence is achieved, the remaining unknowns are solved directly. The efficiency of this technique is illustrated by the numerical computations of two examples previously solved by the cubic spline Galerkin method.     

Simulation of nonlinear boundary conditions by electrical networks

The special features of the method of successive approximations as applied to the simulation of nonlinear boundary conditions by electrical networks are examined. The dependence of the convergence of the method on the initial data is analyzed. Results of an experimental test of the conclusions are presented.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 17, No. 5, pp. 918–925, November, 1969.     

On three-dimensional boundary element methods for magnetostatics invector variables

The boundary-element solution of three-dimensional magnetostatic fields is dealt with. The formulations are based on the magnetic vector potential and the magnetic flux density. The proper boundary-conditions of the problem are discussed, and vector boundary integral equations are presented. An isoparametric boundary element method is used for the solution. Numerical examples are given for both of the formulations     

Finite element formulation of exact non‐reflecting boundary conditions for the time‐dependent wave equation

A modified version of an exact Non‐reflecting Boundary Condition (NRBC) first derived by Grote and Keller is implemented in a finite element formulation for the scalar wave equation. The NRBC annihilate the first N wave harmonics on a spherical truncation boundary, and may be viewed as an extension of the second‐order local boundary condition derived by Bayliss and Turkel. Two alternative finite element formulations are given. In the first, the boundary operator is implemented directly as a ‘natural’ boundary condition in the weak form of the initial–boundary value problem. In the second, the operator is implemented indirectly by introducing auxiliary variables on the truncation boundary. Several versions of implicit and explicit time‐integration schemes are presented for solution of the finite element semidiscrete equations concurrently with the first‐order differential equations associated with the NRBC and an auxiliary variable. Numerical studies are performed to assess the accuracy and convergence properties of the NRBC when implemented in the finite element method. The results demonstrate that the finite element formulation of the (modified) NRBC is remarkably robust, and highly accurate. Copyright © 1999 John Wiley & Sons, Ltd.     

Finite difference approximation of boundary conditions along irregular boundaries

The approximation of normal derivatives along a curved boundary becomes a major difficulty in obtaining finite difference solutions for irregular regions. A number of authors (for-example, Allen,¹ Fox,² Greenspan³ and Parker and Ma⁴) have suggested various solutions to this problem. However, most of these suggested methods are fairly cumbersome to use since they require the use of different formulas for various combinations of mesh and boundary geometries. This note shows a different way to approximate a quite general boundary condition along curved boundaries. The method uses a uniformly spaced mesh, is simple, general, relatively easy to program and appears to give accurate results.     

A boundary element formulation for design sensitivities in materially nonlinear problems

Summary This paper presents a formulation for the determination of design sensitivities for shape optimization in materially nonlinear problems. This approach is based on direct differentiation (DDA) of the relevant boundary element method (BEM) formulation of the problem. It combines the accuracy advantages of the BEM without the difficulty of dealing with strongly singular kernels. This approach provides a new avenue towards efficient shape optimization of small strain elastic-viscoplastic and elastic-plastic problems.With 1 Figure     

New formulation of the transparent boundary conditions for the parabolic equation

New approximate transparent boundary conditions for the nonstationary parabolic (Schrödinger) equation are derived using the method of multiple scales.     

New formulation of the absorbing boundary conditions for the wave equation

New approximate absorbing boundary conditions for the wave equation are derived using the method of multiple scales.     

An implicit corrected SPH formulation for thermal diffusion with linear free surface boundary conditions

The smoothed particle hydrodynamics (SPH) method has proven useful for modeling large deformation of fluids including fluids with stress‐free surfaces. Because of the Lagrangian nature of the method, it is well suited to address the thermal evolution of these free surface flows. Boundary conditions at the interface of the fluid with a solid wall are usually enforced through the use of boundary particles. However, applying conditions at free surfaces, in particular gradient boundary conditions, can be problematic with traditional SPH formulations due to the degradation of the gradient approximation in these regions. Compounding this difficulty is that traditional approximations of the Laplacian operator suffer a similar degradation near free surfaces. A new SPH formulation of the Laplacian operator is presented, which improves the accuracy near free surface boundaries. This new form is based on a gradient approximation commonly used in thermal, viscous, and pressure projection problems, but includes higher‐order terms in the appropriate Taylor series. Comparisons with other approximations of second‐order derivatives are given. The discretization is tested by solving steady‐state and transient problems of thermal diffusion using the Backward Euler method with a GMRES solver. Boundary conditions are imposed through an augmented matrix. Copyright © 2008 John Wiley & Sons, Ltd.     

On the boundary conditions for the vector potential formulation in electrostatics

The vector potential formulation is a promising solution method for nonlinear electromechanically coupled boundary value problems. However, one of the drawbacks of this formulation is the non‐uniqueness of the (electric) vector potential in three dimensions. The present paper focuses on the Coulomb gauging method to overcome this problem. In particular, the corresponding gauging boundary conditions and their consistency with the physical boundary conditions are examined in detail. Furthermore, certain topological features like cavities and multiply connectedness of the domain of analysis are taken into account. Different variational/weak formulations being appropriate for finite element implementation are described. Finally, the suitability of these formulations is demonstrated in several numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

Successive linear approximation for boundary value problems of nonlinear elasticity in relative-descriptional formulation

A method of successive linear approximation in current configuration for solving boundary value problems of large deformation in finite elasticity is proposed. Instead of using Lagrangian or Eulerian formulation, we can also formulated the problem relative to the current configuration, and linearize the constitutive function at the present state so that it leads to a linear boundary value problem for an incremental time step. Therefore, as linearization at present state proceed in time, problem for large deformation can be formulated. The idea is similar to the Euler’s method for differential equations. As an example for the proposed method, numerical simulation of bending a rectangular block into a circular section for Mooney-Rivlin material is given for comparison with the exact solution, which is one of the well-known universal solutions in finite elasticity.     

The wave collapse analyzed for the nonlinear Klein-Gordon equation with periodic boundary conditions

The integral estimation method is used to study the wave collapse dynamics for the Klein-Gordon equation with a nonlinearity of the general form in the case of periodic boundary conditions. Sufficient integral criteria (generalizing the known ones) for the wave collapse are formulated.     

Solution of a nonlinear heat conduction problem with boundary conditions of the fourth kind

We present a method for electrically modelling nonlinear contact heat-transfer problems both with and without taking into account the thermal conductivity of the contact layer.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 18, No. 2, pp. 328–332, February, 1970.     

A unified formulation for vibration analysis of functionally graded shells of revolution with arbitrary boundary conditions

This paper describes a general formulation for free, steady-state and transient vibration analyses of functionally graded shells of revolution subjected to arbitrary boundary conditions. The formulation is derived by means of a modified variational principle in conjunction with a multi-segment partitioning procedure on the basis of the first-order shear deformation shell theory. The material properties of the shells are assumed to vary continuously in the thickness direction according to general four-parameter power-law distributions in terms of volume fractions of the constituents. Fourier series and polynomials are applied to expand the displacements and rotations of each shell segment. The versatility of the formulation is demonstrated through the application of the following polynomials: Chebyshev orthogonal polynomials, Legendre orthogonal polynomials, Hermite orthogonal polynomials and power polynomials. Numerical examples are given for the free vibrations of functionally graded cylindrical, conical and spherical shells with different combinations of free, shear-diaphragm, simply-supported, clamped and elastic-supported boundary conditions. Validity and accuracy of the present formulation are confirmed by comparing the present solutions with the existing results and those obtained from finite element analyses. As to the steady-state and transient vibration analyses, functionally graded conical shells subjected to axisymmetric line force and distributed surface pressure are investigated. The effects of the material power-law distribution, boundary condition and duration of blast loading on the transient responses of the conical shells are also examined.