The MFS for numerical boundary identification in two-dimensional harmonic problems

In this study, we briefly review the applications of the method of fundamental solutions to inverse problems over the last decade. Subsequently, we consider the inverse geometric problem of identifying an unknown part of the boundary of a domain in which the Laplace equation is satisfied. Additional Cauchy data are provided on the known part of the boundary. The method of fundamental solutions is employed in conjunction with regularization in order to obtain a stable solution. Numerical results are presented and discussed.     

Modelling and state estimation for a two-dimensional moving boundary problem

The five-spot waterflooding problem has been modelled as a two-dimensional moving boundary problem with a sharp interface separating the water and oil regions. The Galerkin method was used to solve for the shape and movement of the interface, as well as for the pressures in the reservoir. Having obtained a working model, the extended Kalman filter was then used to estimate the interfacial position when using corrupted pressure measurements from a single sensor in the field.     

Some moving boundary problems for a generalised diffusion equation

Diffusion controlled growth of planar, cylindrical or spherical particles is considered where the composition satisfies a generalised diffusion equation. Asymptotic methods are used to determine composition gradients near the growing particles in the cases of “fast” or “slow” growth. It is shown that in these limits under certain conditions a constant average diffusion coefficient can be defined for which the growth rate calculated from the differential equation with this constant diffusion coefficient agrees with that with the full variable coefficient. Similar considerations are also made of the case of composition dependent diffusion and bulk flow for growing spherical particles.     

Application of schemes on moving grids for numerical simulation of hypervelocity impact problems

We describe a numerical technique for solving hypervelocity impact problems. Computational method is based on Godunov scheme on moving grid. To describe flows with strong deformations a technique of decomposition of numerical region into subregions is developed. The boundaries of subregions can be moved both in Eulerian and Lagrangian fashion. Using the method developed several multimaterial problems with strong deformations have been solved.

To apply Godunov method for elastic-plastic flow conservative form of governing equations is used, which allows one to obtain jump conditions in the case of discontinuous flow.     

A moving Kriging interpolation‐based meshless method for numerical simulation of Kirchhoff plate problems

This paper mainly proposes an alternative way for numerical implementation of thin plates bending based on a new improvement of meshless method, which is combined between the standard element‐free Galerkin method and one different shape functions building technique. The moving Kriging (MK) interpolation is applied instead of the traditional moving least‐square approximation in order to overcome Kronecker's delta property where the standard method does not satisfy. Obviously, the deflection of the thin plates is approximated via the MK interpolation. To illustrate this approach, numerical analysis is examined in both regular and irregular systems. Three examples with different geometric shapes of thin plates undergoing a simply supported boundary are performed. In addition, two important parameters of the present method are also analyzed. A good agreement can be found among the proposed, analytical and finite element methods. Copyright © 2008 John Wiley & Sons, Ltd.     

An approximate method for evaporation problem with mushy zone

In the present paper, a simple mushy zone model is used to track the moving boundaries in an evaporation problem in which the vapor is removed upon formation. Two main parameters for the mushy zone model are analyzed as well as their effect on the movement of the moving boundaries and the thickness of the mushy zone. A new approximate method is developed for analysis and tracking the moving boundaries appears throughout the process. The proposed method mainly based on applying the boundary integral equation corresponding to each phase in such a way that the associated boundary and initial conditions as well as energy equations at the moving boundaries achieved with minimum error and low number of iterations. The results of the present paper seem to be good because there are neither analytical or numerical solutions available.     

A boundary element alternating method for two-dimensional mixed-mode fracture problems

A boundary element alternating method (BEAM) is presented for two dimensional fracture problems. An analytical solution for arbitrary polynomial normal and tangential pressure distributions applied to the crack faces of an embedded crack in an infinite plate is used as the fundamental solution in the alternating method. For the numerical part of the method the boundary element method is used. For problems of edge cracks a technique of utilizing finite elements with BEAM is presented to overcome the inherent singularity in boundary element stress calculation near the boundaries. Several computational aspects that make the algorithm efficient are presented. Finally the BEAM is applied to a variety of two-dimensional crack problems with different configurations and loadings to assess the validity of the method. The method gave accurate stress-intensity factors with minimal computing effort.     

Extended particle difference method for moving boundary problems

In this paper, we present an extended particle difference method for moving boundary (or interface) problems. The particle derivative approximation is further developed to approximate the Taylor polynomial with the moving least squares method through completely node-wise computations. The discontinuity (or singularity) in the derivative field and the kinetic relation of the interfacial physics are effectively immersed into the particle derivative approximation. The segmented points that discretize the moving interface are consecutively updated in an explicit manner to track the evolution of the moving interface topology. Discretized difference equations for the governing equations are directly constructed at the nodes and the segmented points of the computational domain and the moving interface, respectively. Assemblage of the discretized difference equations yields a linear algebraic system of equations that provides efficient and stable solution procedure for a strong formulation. Numerical examples demonstrate that this method achieves excellent accuracy and efficiency for moving boundary problems.     

Some two-dimensional heat-conduction problems with mixed boundary conditions

Successive application of a series of conformal mappings is used to determine the steady-state heat flow and temperature fields in two-dimensional rectangular configurations with mixed boundary conditions. A uniform medium is considered. The analytical solutions are compared with results obtained by means of an electro -hydrodynamic analog.     

A fixed domain method for diffusion with a moving boundary

Summary The one-dimensional diffusion equation for a region with one fixed boundary and one unknown moving boundary is transformed to a non-linear equation on a fixed region by using the moving boundary position as the time variable. The boundary velocity becomes a second dependent variable, with dependence only on the new time variable. An implicit finite difference scheme, marching in time, is applied to a problem with known analytic solution to demonstrate the computing speed and accuracy of this approach, and also to a problem solved previously by variable time step methods. This transformation reduces any parabolic or elliptic system of equations on a domain with moving boundary, or with unknown free surface in two space variables, to a non-linear fixed domain system which has advantages for computation.     

A superelement for two-dimensional singular boundary value problems in linear elasticity

The finite element method for elliptic boundary value problems has been modified to deal with boundary singularities. We introduce a singular-super-element (SSE) which incorporates the known expansion for the singular solution explicitly over the internal region surrounding the singular point, whilst using blended trial functions over the intermediate region, which joins the internal and external regions smoothly. The SSE conforms with the mesh used in the external region, and may be easily incorporated into standard finite element programs. The calculations yield the expansion coefficients directly, as well as an accurate representation of the displacements in the vicinity of the singular point, for a crack or V-notch of any angle subject to any mode of loading. The SSE has been applied to determine stress intensity factors for two-dimensional crack and V-notch problems, including mixed mode. The computations converge rapidly, yielding results of high accuracy.     

A boundary element-free method (BEFM) for two-dimensional potential problems

Combining the boundary integral equation (BIE) method and improved moving least-squares (IMLS) approximation, a direct meshless BIE method, which is called the boundary element-free method (BEFM), for two-dimensional potential problems is discussed in this paper. In the IMLS approximation, the weighted orthogonal functions are used as the basis functions; then the algebra equation system is not ill-conditioned and can be solved without obtaining the inverse matrix. Based on the IMLS approximation and the BIE for two-dimensional potential problems, the formulae of the BEFM are given. The BEFM is a direct numerical method in which the basic unknown quantity is the real solution of the nodal variables, and the boundary conditions can be applied directly and easily; thus, it gives a greater computational precision. Some numerical examples are presented to demonstrate the method.     

Hermitian cubic boundary elements for two-dimensional potential problems

A hermite interpolation based formulation is presented for the boundary element analysis of two-dimensional potential problems. Two three-noded Hermitian Cubic Elements (HCE) are introduced for the modelling of corners or points with non-unique tangents on the boundary. These elements, along with the usual two-noded HCE, are used in numerical examples. The results obtained show that faster convergence can be achieved using HCE compared with using Lagrange interpolation type Quadratic Elements (QE), for about the same amount of computing resources.     

A fast multipole boundary element method for solving two-dimensional thermoelasticity problems

A fast multipole boundary element method (BEM) for solving general uncoupled steady-state thermoelasticity problems in two dimensions is presented in this paper. The fast multipole BEM is developed to handle the thermal term in the thermoelasticity boundary integral equation involving temperature and heat flux distributions on the boundary of the problem domain. Fast multipole expansions, local expansions and related translations for the thermal term are derived using complex variables. Several numerical examples are presented to show the accuracy and effectiveness of the developed fast multipole BEM in calculating the displacement and stress fields for 2-D elastic bodies under various thermal loads, including thin structure domains that are difficult to mesh using the finite element method (FEM). The BEM results using constant elements are found to be accurate compared with the analytical solutions, and the accuracy of the BEM results is found to be comparable to that of the FEM with linear elements. In addition, the BEM offers the ease of use in generating the mesh for a thin structure domain or a domain with complicated geometry, such as a perforated plate with randomly distributed holes for which the FEM fails to provide an adequate mesh. These results clearly demonstrate the potential of the developed fast multipole BEM for solving 2-D thermoelasticity problems.     

Curved composite boundary elements for second-order two-dimensional problems

Finite elements with a curved edge often require relatively large numerical effort to form.^1–4 Relatively simple triangular elements with a single curved edge are developed in this paper for second-order, two-dimensional problems. A convex boundary element is formed as a composite of two straight-edged triangles and a circular sector. Application of the convex composite boundary element results in less numerical effort for a comparable error in circular and elliptical domain test problems than the application of straight-edged elements in all cases shown, and in most cases when compared to the curved isoparametric elements. For domains with concave boundaries, the application of straight-edged, concave composite boundary and curved isoparametric elements give comparable accuracies and numerical efforts because of a fortuitous cancellation of error that occurs with straight-edged elements in this case.     

A numerical method for the solution of two-dimensional inverse heat conduction problems

A numerical method for the solution of inverse heat conduction problems in two-dimensional rectangular domains is established and its performance is demonstrated by computational results. The present method extends Beck's⁸ method to two spatial dimensions and also utilizes future times in order to stabilize the ill-posedness of the underlying problems. The approach relies on a line approximation of the elliptic part of the parabolic differential equation leading to a system of one-dimensional problems which can be decoupled.     

A modified indirect boundary element method for solving two-dimensional sound radiation problems

This paper proposes an engineering remedy to circumvent numerical shortcomings inherent in the indirect boundary element method (IBEM) to two-dimensional sound radiation problems. It is shown that when the acoustic sources are arranged on a shrunk internal boundary, which is produced by a uniform scaling factor applied to the actual vibrating boundary, then fictitious eigenfrequencies `move' to larger values being inversely proportional to this factor. In this way, IBEM analysis becomes capable of treating the problem of the nonuniqueness in a simple and efficient practical manner, which makes the method applicable to praxis. A conservative `a-priori' known scaling factor is established. The proposed method is applied to a circular, a square and rectangular vibrators. Dieser Beitrag beschreibt ein technisches Hilfsmittel, durch dessen Anwendung numerische Schwierigkeiten, die mit dem indirekten Randelementverfahren (IBEM) in zweidimensionalen Schallabstrahlungsproblemen verbunden sind, vermieden werden Können. Es wird hiermit folgendes aufgezeigt. Wenn Schallquellen an einen geschrumpften internen Rand gebracht sind, der entsteht, wenn der real schwingende Rand um einen Stufenfaktor gleichmäßig reduziert wird, dann gehen künstliche Eigenfrequenzen zu höheren Werten, die in umgekehrtem Verhältnis zu diesem Faktor stehen. Dadurch kann die IBEM-Analyse das Problem der nicht Einzigartigkeit auf eine einfache und effiziente Weise behandeln, was die Methode praktisch anwendbar macht. Ein konservativer, von vornherein bekannter Stufenfaktor ist bestimmt. Die vorgeschlagene Methode wird auf kreisförmige, viereckige und rechtwinklige Oszillatoren angewendet.     

A meshless local boundary integral equation method for two-dimensional steady elliptic problems

A novel meshless local boundary integral equation (LBIE) method is proposed for the numerical solution of two-dimensional steady elliptic problems, such as heat conduction, electrostatics or linear elasticity. The domain is discretized by a distribution of boundary and internal nodes. From this nodal points’ cloud a “background” mesh is created by a triangulation algorithm. A local form of the singular boundary integral equation of the conventional boundary elements method is adopted. Its local form is derived by considering a local domain of each node, comprising by the union of neighboring “background” triangles. Therefore, the boundary shape of this local domain is a polygonal closed line. A combination of interpolation schemes is taken into account. Interpolation of boundary unknown field variables is accomplished through boundary elements’ shape functions. On the other hand, the Radial Basis Point Interpolation Functions method is employed for interpolating the unknown interior fields. Essential boundary conditions are imposed directly due to the Kronecker delta-function property of the boundary elements’ interpolation functions. After the numerical evaluation of all boundary integrals, a banded stiffness matrix is constructed, as in the finite elements method. Several potential and elastostatic benchmark problems in two dimensions are solved numerically. The proposed meshless LBIE method is also compared with other numerical methods, in order to demonstrate its efficiency, accuracy and convergence.     

A numerical model for the simulation of two-dimensional convective-dispersion in shallow estuaries

A numerical model for describing two-dimensional convective-dispersive processes in tidally influenced shallow bays in given. The model uses a multi-stage implicit technique with finite difference formulation, for solving the basic mass balance equation for a conservative constituent. The basic inputs to this model consist of instantaneous tidal velocities and depths obtained from an operational tidal hydrodynamic model. The sensitivity of the output concentrations to the variation of dispersion coefficients is investigated. Application of the model to a practical situation is demonstrated by considering Galveston Bay. Texas, U.S.A. The transport model is operated to generate results which are compared with measurements from dye release studies made in a physical model of Galveston Bay.