A finite element–boundary element method for advection–diffusion problems with variable advective fields and infinite domains

SummaryThis paper presents a numerical approach for the design sensitivity analysis of modal frequencies of elastic structures submerged in an unbounded heavy fluid domain. Because the feedback of sound pressure onto the fluid‐loaded structures has to be taken into account in this case, a fully coupled scheme which combines the structural finite element method (FEM) and the acoustic boundary element method (BEM) is employed in the numerical simulation. The resulting nonlinear eigenvalue problem created by the coupled finite element and boundary element (FE‐BE) method is converted into a generalized eigenvalue problem (GEP) of reduced dimension by using a contour integral method. The design sensitivity formulation for non‐repeated eigenvalues is derived based on an adjoint method which uses both the left and right eigenvectors, and a small GEP is formed to calculate the gradients of repeated eigenvalues. The Burton‐Miller formulation is employed to shift the fictitious eigenfrequencies and their derivatives, and the parameter setting which is beneficial to the filtering of fictitious eigenfrequencies is investigated. Numerical examples are given to verify the accuracy and effectiveness of the developed method.

Several analytical techniques were examined for predicting GaAs MMIC (monolithic microwave integrated circuit) junction temperatures to support reliability life tests. Series solution, finite difference, finite element, and boundary element methods were compared using a simple MMIC structure, and the boundary element method was found to be better for predicting MMIC junction temperatures. The series solution method, although it cannot model the plating layers on top of the device, was found to be quite useful for performing preliminary analyses. FETs on several MMIC devices were analyzed using the boundary element method. The thermal effects of several physical features and modeling assumptions were investigated. The thermal model predictions were compared with infrared imager and liquid crystal experimental data.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Application of the trajectory image method to solve convection-diffusion equations

Due to the development of urban areas, underground structures are increasingly needed. Therefore, not only it is likely to have two tunnels located close to each other, but also it happens frequently.Because of the complexity of realistic problems, closed form analytical solutions are not accessible for them. However, latest advances in computational techniques have made numerical approaches more practical for realistic problems. Boundary element method (BEM) is one of such approaches. This technique formulates the problem in terms of boundary values. The main advantage of BEM, especially in comparison with finite element and finite difference methods, is that the discretization is only applied to the boundary, thus reducing the volume of modeling and the number of unknown variables. Moreover, the radiation condition at infinity is exactly satisfied in this method which is very striking for wave propagation problems. First the formulation of boundary element method in the time domain is presented and then a parametric study on the dynamic response of unlined circular tunnels subjected to internal explosion is done. Depth of tunnels, vertical and horizontal distance between two tunnels, diameter of tunnels, and other parameters related to geometry are important factors which can affect displacements and stresses of tunnels under explosive loading. In the present study, effect of two parameters has been investigated: changing the depth of the tunnel that is subjected to internal explosion (case A) and changing the horizontal distance between two shallow tunnels subjected to explosion (case B). Radial displacements and circumferential stresses are the results obtained. Case A- Tunnel depth: The tunnel behavior in depth of H=50R is alike to behavior of same tunnel in infinite depth. Also, radial displacement in upper point of tunnel in depth of H=2R is about 2.7 times of the same tunnel in infinite depth. Circumferential stress in points placed on horizontal diameter of tunnel in depth of H=2R is about 1.8 times of the same tunnel in infinite depth. Case B- Tunnels distance: Two tunnels with horizontal distance of greater than L=50R don’t affect each other. When horizontal distance parameter of two tunnels is L=2.5R then the maximum normalized displacement of the side point of the loaded tunnel, is approximately 2.15 times of the same tunnels with distance of infinite. Also, when horizontal distance of two tunnels is L=2.5R, circumferential stress in lower point of the loaded tunnel, which is normalized with static stress, is approximately 1.2 times of the same tunnels with distance of infinite. Normalized results calculated here display the importance of the geometric parameters on the behavior of unlined tunnels subjected to internal explosion and provide suitable criteria for displacement design of shallow tunnels that are subjected to internal explosion. The accuracy of the present study for the computation of radial displacements and circumferential stresses is verified by solving several problems.

The study of groundwater and porous media flow has become important during the last years in fields as different as civil engineering, aquifer managing and petroleum engineering. The fundamental equations describing these physical phenomena are well known [5], [6] and before the computers allowed them to be solved numerically, many authors developed analytical solutions for special cases [57], [66]. Nowadays, analytical solutions are very important because they allow to check the numerical methods. Different numerical methods have been developed to solve numerically porous media flow problems. The finite difference method (FDM), the finite element method (FEM) and the boundary element method (BEM) have successfully been applied to solve the equations. The application of these three methods to the computation of groundwater flow has shown that the BEM is the most suitable, since it allows the most efficient handling of free surfaces, infinite flow domains and singularities. Besides, the BEM gives more accurate results than the two other methods because the weighting functions used by this method are analytical solutions of the governing equations. In this book, the application of the BEM to the different types of groundwater flows, especially to those described by the Laplace equation, will be discussed.KeywordsPorous MediumFree SurfaceGroundwater FlowBoundary Element MethodRepresentative Elementary VolumeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The boundary element method (BEM) is now a versatile and powerful tool of computational mechanics which has become a popular alternative to the well-established finite element method (FEM). Recently, some authors indicated that the presence of any domain integrals undermines most advantages of the BEM. We believe that this is not true if the domain integral contains known quantities (such as body sources or initial values). The actual sources of advantages of the BEM formulation are not determined by the question of if it is necessary to discretize the domain or not. The BEM formulation for the solution of a boundary value problem exhibits pure boundary character if the solution at any internal point can be expressed in terms of the boundary integrals of relevant quantities and the domain integrals of known body sources and initial values, if any. In other words, the solution can be expressed at an internal point without the need to know the solution at any other internal point. If the boundary values of the relevant quantities were known exactly and the integrations were performed with absolute accuracy, the integral representation would present the exact solution of the boundary value problem. That is why in the BEM solution, extreme emphasis is put on the accuracy of the approximations of boundary values and computation of boundary integrals. Since in the domain discretization techniques (such as FEM, finite difference method, collocation methods) the unknowns at all nodal points are computed simultaneously, the computational error is accumulated.

Computer numerical simulations for accelerated design of rare earth permanent magnet motors in renewable energy applications: A review

A boundary element method for analysing non-linear phenomena on gas-liquid two-phase free surface at middle and higher Reynolds numbers has been developed. Employing this method we analyze two kinds of non-linear phenomena on gas-liquid two-phase free surface (l) Solitary wave on free falling thin liquid film. (2) Surface tension drives convective flows in open-boat type crystal-growth techniques in microgravity environment. INTRODUCTION Although some new schemes of discretizing convective terms are developed as QUICK and QUICK ER in finite difference methods[FDM] and finite element methods[FEM], but it is very difficult to avoid numerical diffusion introduced by convective discretizations, especially, at high Reynolds number. In [1], they developed a boundary element iterative scheme(BEIS) to calculate convective terms and compared the advantages of using upwind or central difference schemes and BEIS. As shown as numerical results, BEIS of the convective terms was the most accurate of the two approaches. However, computation was complex. The main objective of this research is to develop a boundary element method for analysing some non-linear phenomena on gas-liquid two-phase free surface at middle and higher Reynolds numbers. On the basis of the boundary element formulation with the pressure penalty function, the divergence theorem is applied to the non-linear convective volume integral of the boundary element formulation. Consequently, velocity gradients are eliminated, the complete formulation is written in terms of velocity. This avoids the difficulty of convective discretizations, and provides considerable reducing storage and computational requirement while improving accuracy. In order to calculate free surface, we construct shooting iterative method of free surface on the basis of normal stress balance condition Project supported by the National Natural Science Foundation of China Transactions on Modelling and Simulation vol 6, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 168 Free and Moving Boundary Problems similar to [2] [3]. The method is applied to analyze two kinds of free surface problems. (1) Solitary wave on free falling thin liquid film. This problem was researched by Wasden and Kheshgi etc. with finite difference method [4] and finite element method [5]. However, in finite difference method[4], one applied an adjustment procedure for wave velocity in order to obtain reasonable flow fields, in finite element methods[5] numerical solution was limited to low flow rates. Moreover, the upwind schemes of discretizing convective momentum terms common to FDM and FEM introduce numerical diffusion or oscillatory behavior of the wave peak region, more importantly, the upwind scheme lacks sensitivity to cross stream diffusion and source terms which are especially important in the case of a thin film. Therefore it is valuable to research a non-traditional numerical method such as BEM. (2) Surface tension drives convective flows in open-boat type crystal growth techniques in microgravity environment. The analytical or numerical study of free surface problem in crystal-growth techniques is very difficult. We shall restrict ourselves to model a problem of open-boat type crystal growth technology, and we only study the melt-gas free boundary, assume the melt-crystal interface shape is flat. The free surface problem of open-boat type is simplified as the free surface problem in open rectangular cavity. The left vertical boundary of the cavity can be interpreted as the crucible wall. The right vertical boundary of the cavity can be interpreted as the melt-crystal interface. The temperatures equal #/ and 0,., respectively. Some important numerical results are presented in paper. BASIC EQUATIONS We consider a system of steady incompressible viscous liquid in a open rectangular cavity with differentially heated lateral walls (the temperature difference is A0*). There is a gradient of surface tension on the free surface since the distribution of temperature is non-homogeneous. The gradient of surface tension drives the convection of liquid. On the basis of the basic law of fluid mechanics and Boussinesq approximation, the governing equations of these problems can be described by a tensor notation and a non-dimensional form as follow: continuity equation «,-„• = o (i) momentum equations energy equation V,-^,=0 (3) where, #,„% is the Kronecker delta symbol, m = x for solitary wave, m = y for thermocapillary convection. The lengths, velocities are scaled with respect to H, Transactions on Modelling and Simulation vol 6, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X

A Monte Carlo solution method for linear elasticity

A numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.

Holes in the skull may have a large influence on the EEG and ERP. Inverse source modeling techniques such as dipole fitting require an accurate volume conductor model. This model should incorporate holes if present, especially when either a neuronal generator or the electrodes are close to the hole, e.g., in case of a trephine hole in the upper part of the skull. The boundary element method (BEM) is at present the preferred method for inverse computations using a realistic head model, because of its efficiency and availability. Using a simulation approach, we have studied the accuracy of the BEM by comparing it to the analytical solution for a volume conductor without a hole, and to the finite difference method (FDM) for one with a hole. Furthermore, we have evaluated the influence of holes on the results of forward and inverse computations using the BEM. Without a hole and compared to the analytical model, a three-sphere BEM model was accurate up to 5-10%, while the corresponding FDM model had an error <0.5%. In the presence of a hole, the difference between the BEM and the FDM was, on average, 4% (1.3-11.4%). The FDM turned out to be very accurate if no hole is present. We believe that the difference between the BEM and the FDM represents the inaccuracy of the BEM. This inaccuracy in the BEM is very small compared to the effect that holes can have on the scalp potential (up to 450%). In regard to the large influence of holes on forward and inverse computations, we conclude that holes in the skull can be treated reliably by means of the BEM and should be incorporated in forward and inverse modeling.

Finite element method (FEM) is widely used in the modeling of electromagnetic systems and as a reliable method to verify analytical modeling of electromagnetic devices in design step. As the method is based on discretization of domains, the electromagnetic continuity conditions at the boundaries of elements should be confirmed. Electromagnetic continuity conditions in FEM modeling were investigated in this article. It was shown that the continuity of tangential component of magnetic field intensity is not guaranteed at the boundaries of elements. The inability of FEM to establish the continuity condition leads to some virtual surface current densities at the boundaries of adjacent elements which does not exist in reality. The existence of virtual surface currents is proved in an example. Then, with the help of a complicated example, it is shown that large number of elements and, consequently, the large number of boundaries between elements may cause great errors in FEM postprocess results, especially in calculating magnetic stored energy and electromagnetic forces acting on moving parts. It has been shown that it is very important to take into account the impact of these virtual current densities in energy conversion analysis by FEM.

Adaptive gridding methods are of fundamental importance both for industry and academia. As one of the computing methods, the Boundary Element Method (BEM) is used to simulate problems whose fundamental solutions are available. The method is usually characterised as constant elements BEM or linear elements BEM depending on the type of interpolation used at the elements. Its popularity is steadily growing because of its advantages over other numerical methods most important of which being that it only involves obtaining data at the boundary and computation of the solution in the domain is merely a case of post processing through the use of an identity. This results in the reduction of the problem dimension by unity. Although there is a reduction in dimension when we use BEM, the method usually results in full matrices which can be expensive to solve. This makes the method costly for problems that require very fine grids like those with hot spots as in the example of impressed current cathodic protection systems (ICCP). The BEM is a global method in nature in that the solution in one node depends on the solutions in all the other nodes of the grid. Hence an error in one node can pollute the solution in all the other nodes. In this thesis, we first focus our attention on defining and studying both the local and global errors for the BEM. This is not a completely new study as the literature suggests, however, our approach is different. We use the basic foundations of the the method to define the errors. Since the method is a global method, first we use the interpolation error on each element to define what we have called a sublocal error. Then using the sublocal error we have defined the local error. Understanding the local errors enabled us study the global error. Theoretical and numerical results show that these errors are second order in grid size for both the constant and linear element cases. Then, having explored errors, we study a method for adaptive grid refinement for the BEM. Rather than using a truly nonuniform grid, we present a method called local defect correction (LDC) that is based on local uniform grid refinement. This method is already developed and documented for other numerical methods such as finite difference and finite volume methods but not for BEM. In the LDC method, the discretization on a composite grid is based on a combination of standard discretizations on several uniform grids with different grid sizes that cover different parts of the domain. At least one grid, the uniform global coarse grid, should cover the entire boundary. The size of the global coarse grid is chosen in agreement with the relatively smooth behaviour of the solution outside the hot spots. Then several uniform local fine grids each of which covers only a (small) part of the boundary are used in the hot spots. The grid sizes of the local grids are chosen in agreement with the behaviour of the continuous solution in that part of the boundary. The LDC method is an iterative process whereby a basic global discretisation is improved by local discretisations defined in subdomains. The update of the coarse grid solution is achieved by adding a defect correction term to the right hand side of the coarse grid problem. At each iteration step, the process yields a discrete approximation of the continuous solution on the corresponding composite grid. We have shown how this discretisation can be achieved for the BEM. We apply the discretisation to an academic example to demonstrate its implementation and later show how to use it for an application as the ICCP system. The results show that it is a cheaper method than solving on a truly composite grid and converges in a single step.

The article considers one of the methods of determining the reduced resistance of heat transfer in architectural and structural nodes and zones of heat-conducting inclusions of enclosing structures for their further optimization according to the principle of the maximum possible reduction of heat loss. The corresponding algorithm is studied, which involves a cyclical passage of several stages: analysis of the current state/design of the nodal solution by the method of finding the trajectories of the fastest heat loss (like modeling the lines of the fastest ascent or descent using a one-dimensional search) and determining the heat transfer resistances along the corresponding trajectories, based on which the calculation the value of the reduced heat transfer resistance. The process of searching for the trajectories of the fastest heat energy leaks is implemented by means of geometric modeling of special interpolation functions built on reference values ​​of the temperature fields in the studied nodes. Based on the results of the analysis of the obtained trajectories of the fastest thermal energy leaks, it is proposed to rationalize nodal design solutions by making constructive changes along the path of simulated heat loss trajectories (thermal bridges or cold bridges) and checking the effectiveness of the measures taken by recalculating and comparing the calculated reduced heat transfer resistances with normative indicators. taking into account regulatory requirements regarding the specific energy consumption of buildings. Temperature fields are reproduced by one of the methods of numerical modeling (finite element method, finite difference method or boundary element method), for which it is permissible to use specialized software, or using integral formulas to determine temperature indicators in the studied architectural and structural nodes. To construct special interpolation functions of the temperature field, it is suggested to use one of the most widely used approaches based on Lagrange or Newton interpolation polynomials. Interpolation based on radial basis functions is also acceptable for the functional description of a discrete series of points on the plane. The detection and localization of thermal bridges in the architectural and structural nodes of the enclosing structure is one of the most important tasks of building physics in the field of increasing the energy efficiency of buildings and, as a result, reducing their carbon footprint.

The boundary element method (BEM) is an efficient numerical method that approximates solutions of various boundary value problems. Despite its success little research has been performed on the conditioning of the linear systems that appear in the BEM. For a Laplace equation with Dirichlet boundary conditions a remarkable phenomenon is observed; the corresponding boundary integral equation (BIE) is singular for a certain critical size of the 2D domain. As a consequence the discrete counterpart of the BIE, the linear system, is singular too, or at least ill-conditioned. This is reflected by the condition number of the system matrix, which is infinitely large, or at least very large. When the condition number of the BEM-matrix is large, the linear system is difficult to solve and the solution of the system is very sensible to perturbations in the boundary data. For a Laplace equation with mixed boundary conditions a similar phenomenon is observed. The corresponding BEM-matrix consists of two blocks; one block originates from the BEM-matrix belonging to the Dirichlet problem, the other block originates from the BEM-matrix belonging to the Neumann problem. The composite matrix inherits the solvability problems from the Dirichlet block. In other words, for the Laplace equation with mixed boundary conditions there exists also a critical size of the 2D domain for which the BEM-matrix has an infinitely large condition number. Hence the size and shape of the domain affects the solvability of the BEM problem. The critical size of the domain for which the BIE becomes singular is related to the logarithmic capacity of the domain. The logarithmic capacity is a positive real number that is a function of the size and shape of the domain. If this logarithmic capacity is equal to one, the domain is a critical domain, and for this domain the BIE becomes singular. Thus by computing the logarithmic capacity we can a-priori determine whether the BIE will be singular or not. The logarithmic capacity depends linearly on the scale of the domain, and thus a domain with logarithmic capacity equal to one can always be found by rescaling the domain. Unfortunately the logarithmic capacity can only be computed analytically for a few simple domains; for more involved domains the logarithmic capacity can be estimated though. There are several possibilities to avoid large condition numbers, i.e. singular BIEs that appear at critical domains. The first option is to rescale the domain such that the logarithmic capacity is unequal to one. One can also add a supplementary condition to the BIE and the linear system. A drawback of this option is that the linear system has more equations than unknowns and different techniques are required to solve the system. A third option is to slightly modify the fundamental solution of the Laplace operator. This fundamental solution directly appears in the BIE and it can be shown that a suitable modification yields BIEs that do not become singular. The critical domains for which the BIEs become singular do not restrict to Laplace equations only. Also for BIEs applied to the biharmonic equation or the elastostatic equations and the Stokes equations such critical domains exist. As the last two equations are vectorial equations, also the corresponding BIE consists of two equations. As a consequence two critical domains can be found for which these BIEs become singular. To obtain nonsingular BIEs techniques similar to the Laplace case can be used. Unfortunately we cannot a-priori determine the sizes for which the BIEs becomes singular, and thus do not know to what size we should rescale the domain to obtain nonsingular BIEs. The existence of critical domains is in essence caused by the logarithmic term in the fundamental solutions for the elliptic boundary value problems in 2D. This logarithmic term does not depend linearly on the size of the domain. When a domain is scaled, i.e. multiplied by a scale factor, the argument of the logarithm is also multiplied by this scale factor, but the logarithm turns this into an additive term. Thus the logarithm transforms multiplication into addition. This affects the BIEs in such a way that critical domains can appear. The fundamental solutions of boundary value problems in 3D do not contain a logarithmic term. Hence scaling of the domain does not affect the fundamental solution, and consequently also the BIE is not affected. Hence we may safely rescale 3D domains without the risk to encounter a critical domain. An example in which a domain takes many different sizes and shapes is the blowing problem. In this problem a viscous fluid is blown to a desired shape. Typically the time is discretised into a set of discrete time steps, and at each step the shape of the fluid is computed by solving the Stokes equations. When attempting to simulate this problem in 2D, we meet a large number of 2D domains, and we risk that one of these domains is equal to or approaches a critical domain. In such a case the BEM will have difficulties with solving the Stokes equations for that particular domain. When simulating the blowing problem in 3D, no critical domains are encountered. It turns out that the BEM is a very efficient numerical method for this particular 3D problem with a moving boundary. As we are merely interested in the shape of the fluid, we only need to know the flow of its boundary. The BEM does exactly that; it does not compute the flow at the interior of the fluid. Furthermore it is rather easy to include other effects from the blowing problem in the model, such as gravity, surface tension and friction from the contact of the fluid with a wall. As only the boundary of the fluid is discretised, the system matrices that appear in the BEM are smaller than the system matrices that appear when solving the problem with a finite element method, for example. Though the BEM-matrices are dense, while the finite element matrices are sparse, the computational effort for the BEM is relatively low. In short, the BEMis a very appropriate numerical method when solving blowing problems.