12th GAMM-Seminar Kiel
Information about 12th GAMM-Seminar Kiel
Mathematisches Seminar Bereich 2
Christian-Albrechts-Universität zu Kiel
12th GAMM-Seminar Kiel Information about 12th GAMM-Seminar Kiel
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The GAMM Committee Efficient numerical methods for partial differential equations in cooperation with the Christian-Albrechts-Universität Kiel organizes the
->[Friday] ->[Saturday] ->[Sunday]
9.00 Opening
9.15 - 9.55 R. Schneider (Darmstadt):
Multi-Scale Methods for Boundary Integral Equations
Coffee-break
10.30 - 10.55 B.H. Kleemann (Berlin):
Wavelet method for a logarithmic singular integral equation
arising in scattering
11.00 - 11.30 W.S.Hall, R.A.McKenzie (Middlesbrough, U.K.):
A Study of Multi-Wavelet Transforms for a Non-Conforming
Boundary Element Formulation
11.35 - 12.00 A. Rathsfeld (Berlin):
A wavelet algorithm for the BEM corresponding to the fixed
boundary value problem of geodesy
Lunch
14.30 - 15.00 O. Steinbach (Stuttgart):
Fast Solvers for Boundary Element Methods:
Parallelization and Preconditioning
15.05 - 15.15 R. Lehmann (Karlsruhe), R. Klees (Delft, The Netherlands):
Parallel Setup for Galerkin equation system in a
BEM-solution for a geodetic BVP
15.20 - 15.50 A. Greenbaum, A. Mayo, V. Sonnad (Austin, U.S.A.):
Rapid, Parallel Evaluation of Integrals in Potential
Theory on General Three Dimensional Regions
Coffee-break
16.15 - 16.45 M. Kuhn (Linz, Austria):
Parallel Solution of DD-BEM equations using local
Multigrid Preconditioners
16.50 - 17.15 S.A. Funken, E.P. Stephan (Hannover):
Fast Solvers for Adaptive FEM--BEM Coupling
17.20 - 17.50 K.Türke, E. Schnack (Karlsruhe):
A Two Grid Method for Coupling FEM and BEM in Elasticity
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9.00 - 9.30 S.A. Sauter (Kiel):
Fast Algorithms for Galerkin BEM:
Panel-Clustering and Cubature
9.30 - 10.00 Ch. Lage (Kiel):
Object-oriented design aspects for BEM
Coffee-break
10.30 - 10.55 K. Giebermann (Karlsruhe):
On the Panel Clustering Method for the Helmholtz equation
11.00 - 11.30 C. Gaspar (Gyor, Hungary):
Multigrid and Multipole Techniques in the
Boundary Integral Equation Method
11.35 - 12.05 Y. Yamada (Kyoto, Japan), K. Hayami (Tokyo, Japan):
A multipole boundary element method for two dimensional
elastostatics
Lunch
14.45 - 15.15 B. Faermann (Kiel):
Local a-posteriori error estimators for the discretization
of boundary integral equations
15.20 - 15.40 R. Hochmuth (Berlin):
A-posteriori Estimates for Boundary Elements
Coffee-break
16.20 - 16.50 J.O. Nygaard, J. Grue, H.P. Langtangen, K. Mørken (Oslo, Norway):
On adaptive spline and wavelet methods
for an integral formulation of inviscid flow
16.55 - 17.35 W.L. Wendland (Stuttgart):
An Extraction and Window Technique for BEM
17.45 - ... Poster session, Short presentations will be given by:
- Ke Chen (Liverpool):
Preconditioning Boundary Element Equations
- T. Finck (Chemnitz):
Spline approximation methods for a class of singular
integral equations over plane domains over plane domains
- J.-P. Mayer (Kiel):
Solution to Geodetic Problems by Using the Double-Layer
Potential
- S. Szikrai (Karlsruhe):
A parallel, iterative Neumann solver for
three dimensional elastostatic problems
18.30 Reception
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->[Friday]
->[Sunday]
9.00 - 9.30 Ch. Schwab (Zürich):
Quadrature error analysis for hp-Galerkin BEM on polyhedra
9.35 - 10.00 H. Schippers, F.P. Grooteman (Amsterdam, The Netherlands):
A symmetrical boundary element formulation through cabin walls
Coffee-break
10.20 - 10.50 H. Andrä (Karlsruhe):
A Galerkin-Type Boundary Element Implementation for
3D Elasticity Problems by Using a Computer Algebra System
10.50 - 11.20 Ken Hayami (Tokyo, Japan):
In search of an optimum variable transformation for
nearly singular integrals in BEM
End of 12th GAMM-Seminar Kiel
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->[Friday]
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A Galerkin approximation of both strongly and hypersingular boundary
integral equation (BIE) is considered for the solution of a mixed
boundary value problem in 3D elasticity leading to a symmetric system
of linear equations.
The evaluation of Cauchy principal values (v.p.) and finite parts (p.f.)
of double integrals is one of the most difficult parts within the
implementation of such boundary element methods (BEMs).
A new integration method, which is strictly derived for the cases
of coincident elements as well as edge-adjacent and vertex-adjacent elements,
leads to explicitly given regular
integrand functions which can be integrated by the standard
Gauss-Legendre and Gauss-Jacobi quadrature rules.
Problems of a wide range of integral kernels
on curved surfaces can be treated by this integration method.
We give estimates of the quadrature errors of the singular four-dimensional
integrals.
Because the effort for the analytical evaluation of the integrands of the
element stiffness matrices increases with the complexity of the kernel
functions and with the polynomial order of the shape functions, the
computer algebra system Maple V [MAPLE is
a registered trademark of Waterloo Maple Software] was employed for this task.
A Maple V program greatly simplifies the
implementation of this new integration method especially
for matrix-valued kernel
functions. The computer algebra program produces C or
F77 source code
for the computation of the integrand functions
which can be integrated by standard quadrature rules.
Numerical aspects are discussed.
->[Timetable] ->[GAMM-Homepage]
Boundary element equations are linear systems of equations Ax=b with full and dense matrices A. Although wavelet bases may give rise to nearly sparse matrices, we here consider the commonly used boundary element methods based on collocation with polynomial functions or on quadrature.
Many preconditioning techniques have been proposed in the literature for full linear systems, though some are designed with heuristic assumptions or on sparity consideration only. We investigate a class of such preconditioners that may be applied for a fast solution of singular and weakly-singular boundary integral equations. Our results, both theoretical and numerical, show that those preconditioners that contain the essential singularity of a singular operator or the dominant part of a weakly-singular operator are more efficient and reliable. Examples of an exterior Neumann problem and some integral equations with Cauchy singularity are illustrated.
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Adaptive methods are already well established in the finite element field.
In boundary element methods, on the other hand, only a few theoretical
and numerical results concerning adaptive procedures exist.
The refinement process in adaptive methods is generally controlled by local
a-posteriori error estimators. We have shown that some special local
a-posteriori error estimators of Babuska and Rheinboldt
--which were developed for finite element methods--
are also applicable
to boundary element methods. They are applicable to the Galerkin method, if the
boundary integral operator has an order
alpha in (-1/2,infty) and they are also applicable
to the nodal collocation method with odd degree splines.
The Babuska-Rheinboldt error estimators are bounded by local weighted
Sobolev-Slobodeckij norms of the residual. These computable local values of the
residual are used to control the mesh refinement process.
->[Timetable] ->[GAMM-Homepage]
We consider operator equations $(aI+bS_{G})u=f$ in the $L^2(G)$--space, where
(S_Gu)(x)=-\frac{1}{\pi}\int\limits_G \frac{u(y)}{(x-y)^2}dy
is a Cauchy--type operator over a plane domain $G$ and the coefficients
$a$ and $b$ are continuous functions on the closure $\bar{G}$.
In his book "'The Method of singular integral equations"', A.D. Dzuraev
pointed out the connection between the singular integral operator and certain
boundary value problems.
We present a uniform Banach algebra approach to the stability problem for a bulk
of special approximation methods (including Galerkin and Collocation methods).
This approach is based on non--commutative Gelfand theories (local principles).
We therefore interprete the stability problem as invertibility problem in
suitable constructed Banach algebras.
We prove that those approximation method is stable if and only if a certain
class of discrete convolution operators and Toeplitz
operators on a half plane is invertible. We are able to solve the
invertibility problem for
the corresponding Toeplitz operators, which are generated by non--continuous
functions, up to now for some simple spline functions only. In this case we
formulate the necessary and sufficient stability conditions in terms of
the coefficients $a$ and $b$ of the considered operator.->[Timetable] ->[GAMM-Homepage]
We present two nearly optimal preconditioned iterative methods to solve indefinite
linear systems of equations arising from h-adaptive procedures for the symmetric
coupling
nof Finite Elements and Boundary Elements. These solvers are nearly optimal in the
sense, that their convergence rate grows only logarithmically with the number of
unknowns
They are based either on the conjugate residual method with block diagonal
preconditioning,
where no Schur Complement construction is required, or on an inner--outer iteration
of Axelsson and Vassilevski. Both methods use multilevel additive Schwarz method
of seperate positive semi--definite and negative definite parts of the coupled
operator.
The efficiency of the solvers is shown by numerical experiments yielding
fast convergence.
->[Timetable] ->[GAMM-Homepage]
The computational cost of the traditional version of the Boundary Integral Equation Method is O(N^3) where N is the number of boundary nodes. This is due to the relatively bad properties of the boundary element matrices, since they are generally neither self-adjoint nor sparse. In addition to it, if the original differential equation is supplied with mixed boundary condition, the corresponding boundary integral equation is not of the second kind, so that the traditional well-known iterative techniques can hardly be applied. We present a method which shows some similarities to the multiplicative Schwarz method defined along the boundary. In each iteration step, the boundary integral equations of certain pure Dirichlet and Neumann subproblems are to be solved (even if the original boundary condition is of mixed type), which allows the use of standard multigrid tools. We give a theoretical foundation of the method in Sobolev spaces. Convergence theorems as well as numerical examples are presented, which show that the number of the necessary arithmetic operations can be reduced from O(N^3) to O(N^2). Moreover, we also derive a multipole-based technique to evaluate the appearing boundary integrals in an economic way: the overall computational cost can thus be reduced further to O(NlogN) only.
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We present an implementation of the panel clustering method
for the Helmholtz equation, which occurs for example in acoustic scattering
theory.
Using the ansatz of Brakhage and Werner, i.e., representing the solution
as a combination of acoustic single and double layer potential leads to
a Fredholm integral equation of the second kind.
This approach avoids problem with wavenumbers close to eigenfrequencys of
the Laplace operator.
We show the influence of the coupling ansatz to the linear system of
equations resulting from disretization procedure, i.e., from the collocation
or Galerkin method.
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On the Panel Clustering Method for the Helmholtz equation
Klaus Giebermann
Universität Karlsruhe, Institut f. Praktische Mathematik
Englerstr. 2, 76128 Karlsruhe
We also describe a method for the fast computation of a wavelet based
approximation of the collocation or Galerkin matrix using the panel
clustering method.
In search of an optimum variable transformation
Ken Hayami
for nearly singular integrals in BEM
Department of Mathematical Engineering and Information Physics,
University of Tokyo, Japan,
e--mail:
hayami@simplex.t.u-tokyo.ac.jp
The author proposed a variable transformation method for the accurate
calculation of nearly singular integrals over curved surface elements, which
occur in the three dimensional boundary element method when the source point
is very near the element [1].
Taking, as an example, a curved quadrilateral element S described by
x(eta_1,eta_2), the method first finds the closest point
x(overline{eta}_1,overline{eta}_2) on S to the source point
x_s. Then, it calculates the point
tilde{x}_s = tilde{x}(overline{eta}_1,overline{eta}_2) on the
bilinear element tilde{S} defined by the four corner nodes of S. Next,
each triangle tilde{triangle}_j, (j=1,...,4) formed by tilde{x}_s
and two adjacent corner nodes of the element S, is linearly mapped to the
corresponding triangle in the parametric space (eta_1,eta_2). Then,
polar coordinates (rho,theta) centred at tilde{x}_s are introduced
in each triangle tilde{triangle}_j. Next, a radial variable transformation
such as R(rho)=log(rho+d) is introduced in order to weaken the near
singularity of the integrand when the source point is near the element, where
d is the distance between x_s and S. Also, an angular variable
transformation t(theta) is employed to weaken the angular near singularity
when the triangle tilde{triangle}_j is thin. Finally, numerical integration
is performed in the transformed variables R and t.
In the talk, we will first review the method and its theoretical
error estimates, and then introduce some new developments including automatic
numerical integration using the trapezium rule and an attempt to further
optimize the radial variable transformation for the Gauss-Legendre rule.
[1] K. Hayami, A Projection Transformation Method for Nearly Singular Surface Boundary Element Integrals, Lecture Notes in Engineering, Vol.73, Springer-Verlag, 1992.
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We will present a multipole boundary element method (MBEM) for two
dimensional elastostatics. Unlike the biharmonic equation formulation by
Greenbaum et al. [1], we give a direct formulation in terms
of displacement and traction variables, which seems more convenient for
general boundary conditions and applications.
MBEM formulations are given for the Dirichlet, Neumann and mixed boundary
value problems, including the evaluation of the internal stress. At present,
the implemented algorithm requires O(N log N) computational work and
memory, where N is the number of boundary elements. Theoretical error estimates
for the MBEM are also derived.
Numerical examples for Dirichlet, Neumann and mixed boundary problems are
given. The GCR and Bi-CGSTAB were used as the iterative solvers for the MBEM.
The MBEM was compared with the conventional BEM using the LU decomposition or
the same iterative solvers. For the Dirichlet problem, the MBEM with GCR was
the most efficient in CPU time for N>300. However, for the Neumann and mixed
boundary value problems, the MBEM could not compete in CPU-time, since more
iterations were required for convergence. Simple preconditioners such as
diagonal scaling and ILU decomposition only gave minor improvements. The MBEM
was still the most efficient in terms of elapsed time, which is governed by the
memory requirement.
[1] Greenbaum, A., Greengard, L. and Mayo A., On the numerical solution of the biharmonic equation in the plane, Physica D, Vol.60, pp.216-225, 1992.
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The understanding of a-posteriori error estimates for boundary integral methods appears to be by far less developed than for differential equations. The main reason is that unlike e.g. partial differential operators integral operators possess only pseudolocal properties. The objective of this lecture is to consider a-posteriori estimates in the context of multiscale bases oriented methods. We discuss some basic concepts and ideas from this point of view and relate them to existing techniques in more conventional settings. Our local error estimates arise in a fairly unified fashion essentialy as coefficients of corresponding multiscale expansions. In this way we obtain residual based a-posteriori estimates which are reliable and efficient. A concrete adaptive strategy based on these estimates can be proved to converge. In principle, the results apply to a wide class of elliptic problems covering also operators of negative order.
->[Timetable] ->[GAMM-Homepage]
3D scattering at cylindrical objects leads e.g. to the exterior Dirichlet problem for the Helmholtz equation in 2D. The problem is transformed into an equivalent boundary integral equation with logarithmic kernel function. For this equation a fully discrete collocation method with biorthogonal wavelets is presented. The arising stiffness matrix is known to be sparse in the wavelet basis. The number of nonzero entries then is of the order O(N logN) with N the number of unknowns. Using an a-priori compression criterion only these entries are necessary to calculate if the stiffness matrix is assembled directly in the wavelet basis. From this a computational amount of the same order does not directly follow because of the necessity of highly accurate quadrature of the arising integrals if the support of the wavelet is near to the support of the test functional. Therefore great attention has to be put to an efficient quadrature within the process of direct matrix assemblation. The rules are to be adapted to the logarithmic singular kernel and to the wavelet structure. The arising sparse linear system is solved iteratively by the Krylov-subspace method GMRES. Because of the logarithmic singularity the condition number grows with N. Therefore a special preconditioner is applied to get a constant iteration count. Results achieved for the accuracy, convergence, compression and time complexity of the method for various scattering objects and wavenumbers are discussed and compared with theoretical ones from the wavelet theory for pseudodifferential equations.
->[Timetable] ->[GAMM-Homepage]
Software design for the BEM has to encounter several tasks each with a great variety of parameters. Treating these parameters, e.g. discretization schemes or cubature techniques, too restrictive means developing software adequate only for a few applications, i.e., the reusability of the software or even parts of the software is very limited. To circumvent this nuisance one has to isolate the essential concepts of the BEM as well as their interactions to specify an appropriate and flexible design.
We present an object-oriented approach considering these design rules. The identification of key abstractions like geometry, spaces, dualforms, operators and functions is discussed as well as the extensibility of the developed system to advanced methods, especially the panel clustering algorithm.
->[Timetable] ->[GAMM-Homepage]
A particular problem in setting up large BEM equation systems is the huge number of offdiagonal elements, which in Galerkin method are mainly regular integrals over pairs of boundary elements. The numerical effort of cubature depends strongly on the distance between these elements. Therefore, the polynomial degree of exactness of the cubature formula may be choosen based on an estimate of this distance. In a straightforeward implementation on a parallel computer, this may lead to severe imbalances of workload, which drastically reduces the speedup. To overcome this difficulty, dynamic load balancing schemes are applied. Also, different distributions of boundary elements in the parallel computing environment must be considered, which is equivalent to a static balance of workload. It is shown how this works in the solution of the classical oblique BVP of potential theory (linearized version of the fixed gravimetric BVP) on a IBM 9076 SP/2.
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One problem of physical geodesy is the determination of the shape of the Earth and its gravity field from astrogeodetic and gravimetric data. The mathematical formulation of this problem results in a free boundary value problem (the so-called Molodensky problem) or a mixed, i.e. partly free, boundary value problem (the altimetry-gravimetry problem) for exterior domains. We apply the integral equation method using the double-layer ansatz. Let $ D_\varphi^+ (f)$ be the extension onto the boundary of the double-layer potential from the exterior domain. In the same manner we define $\frac{\partial}{\partial l} D_\varphi^+ ( f)$ as the extension of the oblique derivative of the double-layer potential. Here $\varphi$ is a Lipschitz surface and $f$ a density. It is proved that $\frac{\partial}{\partial l} D_\varphi^+ $ is invertible in $W_2^{\frac{1}{2}} / \rset $, provided the direction l(x) is not tangential to the surface. For given $G,H: \SS ^2 \longrightarrow \rset $ we consider the problem of seeking a surface $\varphi$ as the solution of
(*) D_\varphi^+ \left( \left( \frac{\partial}{\partial l}
D_\varphi^+ \right)^{-1} (H) \right) \ = \ G \qquad .
Knowing that this equation has one and only one solution $\varphi$,
we get existence and uniqueness of the linearized Molodensky problem,
too. In order to prove local existence and uniqueness of (*), we apply
the implicit function theorem. In this light we evaluate the
Fr\'{e}chet derivative of (*) w.r.t. surfaces at the sphere. If
the arising operator is invertible we can apply
the implicit function theorem to get local existence and uniqueness.
In this context local means the closeness of the Earth's shape to the
sphere. It is planed to use formulation (*) also for constructing an
iterative procedure to solve the problem.->[Timetable] ->[GAMM-Homepage]
We have constructed a non-conforming Boundary Element formulation for 2D Potential problems. The geometry is approximated by quadratic shape functions and the potential and flux are approximated by Legendre polynomials up to degree two. The purpose of which is to produce a linear system of equations which are appropriate for the the application of the Discrete Multi-Wavelet Transform. The solutions of known problems are compared with the approximations produced by this method once truncation has been applied to the transformed matrix. Plots of accuracy against the sparsity are produced showing the trade off between speed of obtaining the solution and the accuracy. The Schultz method is used to solve the system of equations. This method is efficient when the sparsity of the system matrix is $O(n)$. Higher order approximations are thus considered to keep the matrices sparse enough so the Schultz method remains efficient. The benefits of this wavelet based method are compared to more conventional solution procedures. Extensions and generalisations of the work will be discussed.
->[Timetable] ->[GAMM-Homepage]
The Domain Decomposition Method (DD) is a powerful tool for establishing
weak formulations and for constructing the corresponding parallel solvers.
Although the method allows the coupling of different discretization
techniques,
i.e. Boundary Element Methods (BEM) and Finite Element Methods,
as it is desired in various applications, we restrict
ourselves to pure BEM formulations.
In the talk, we give a brief introduction to the preprocessing tools
which perform an automatic decomposition of the domain of interest into
$p$~subdomains, where
$p$ is the number of processors to be used.
Furthermore, the parallel algorithm and, in particular, the preconditioners
being involved
will be discussed in detail. The latter are
required for the Schur--complement system and for the single layer potential
operator and both are based on local multigrid methods.
Numerical examples, including potential and
linear elasticity problems, which demonstrate the high
efficiency of the algorithm will be presented.
->[Timetable] ->[GAMM-Homepage]
Traditionally, integral equations have been solved numerically by
boundary element methods where the solution is expanded in a sum of
lower order piecewise polynomial basis functions along the boundary.
It is well known that higher order expansions lead to a faster
convergence rate, and hence the possibility of a significant reduction
of the computational work in order to reach a particular level of
accuracy.
In this paper, higher order boundary element methods based on
B-splines and wavelets are described and compared. By numerical
examples, it is shown that the process of choosing a good knot vector
for the spline case corresponds to the compression of the wavelet
matrix. The wavelet method therefore eliminates the possibly tricky
procedure of choosing knot vectors for the splines. On the other hand,
there are limits to how much the matrix of the wavelet method can be
compressed.
The two-dimensional problem is a natural starting point for a thorough
comparison, because it involves all the principal mathematical and
numerical difficulties. The ultimate aim is to develop better methods
for the three-dimensional case.
->[Timetable] ->[GAMM-Homepage]
One of the fundamental problems of geodesy consists in the determination
of the potential field of gravity from the gravity data measured on
the surface of the earth. With the help of linearization and boundary element
techniques the problem can be reduced to the numerical solution of the
singular integral equation
Here n(x) stands for the normal at the point x of the surface of
the earth
Gamma, l is the direction of the reference gravity and the right-hand
side delta g is a certain function of the reference gravity and the
measured gravity. The solution f is the unknown density function of the
single layer representation for the unknown difference gravity potential.
Equation ([?]) can be solved numerically by e.g. a piecewise
bilinear collocation method.
For an accurate approximate solution a high resolution of the geometry
(part of the surface of the earth) is necessary. This leads to large and dense
linear systems. To save storage and computation time we have implemented
a wavelet algorithm for the solution of the collocation system.
We will present and discuss numerical experiments with this algorithm.
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In our talk we will present efficient quadrature strategies for the approximation
of the matrix elements which can be applied to any kind of kernel functions,
surface approximation techniques, and shape functions. Only the
subroutines which evaluates the kernel functions and the shape functions
have to be modified. Furthermore this integration technique is relatively easy to
implement since it can be tested isolated on simple examples.
In the second part of the talk we will formulate the Panel-Clustering method for
the Galerkin BEM in a way which is easy to implement. Only the fundamental
solution which does not depend on possibly complicated surfaces has to be expanded.
The expansions of the kernel functions which are derivatives of the fundamental
solution can be derived very fast from that expansion.
->[Timetable]
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Investigations on the acoustic field around and inside commercial aircraft
are motivated by the urge to reduce cabin noise inside the aircraft. The
exterior acoustic field causes vibrations of the cabin wall, which contri-
bute to the interior noise. The cabin noise inside the aircraft can be
reduced by putting porous acoustically absorbing material (insulation)
just on the inside of the skin panel of the cabin wall.
[1] SCHMIDT, G.: Boundary element discretization of Poincare-Steklov
operators, Numer. Math., Vol. 69, 1994, pp. 83-101.
->[Timetable]
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Boundary element methods offer an appropriate tool for the
numerical solution of certain boundary value problems
in engineering.
A major drawback of BEM is the fact that
the arising system matrices are densely populated,
which is limiting the discretization
of realistic 3D problems with complex geometry.
Like panel-clustering and multi-pole expansion,
biorthogonal wavelet bases remedy this situation by
approximating the discrete scheme in an efficient way.
Multi-scale methods achieve this
by approximating the system matrix relative
to a biorthogonal wavelet basis by a sparse matrix.
We propose a boundary element discretization
for static or low frequency problems
which is based on parametric
surface representation. The surfaces are assumed to be
piecewise analytic and parametrized over
quadrilateral surface (macro-)elements.
Trial functions are supposed to be globally continuous
and piecewise bilinear
in each parameter domain.
We construct a biorthogonal wavelet basis with
desired vanishing moments on each surface (macro-)element.
In addition, this basis satisfy a stability
condition so that the Galerkin wavelet method can be
immediately preconditioned. The proposed basis
have all desired properties achieving
an asymptotically optimal balance between
accuracy and efficiency. This means that we
directly compute a sparse approximation of the system matrix
causing an additional error proportional the optimal error bound
of the Galerkin discretization. This can be done
such that the total number of nonzero
matrix coefficients increases
only linearly with the total number of unknowns $N$.
In order to compute the nonzero matrix coefficients directly,
we propose an adaptive quadrature method,
with the desired accuracy
requiring totally O (N) floating point operations.
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We analyze the quadrature error for a hp-Galerkin BEM on
curved polyhedra. The method is based on subspaces of
high degree polynomials combined with geometric mesh
refinement towards the edges of the domain.
Under suitable regularity assumptions on the solution
generally believed to be valid (and proved in certain
special cases by ELSCHNER and STEPHAN and coworkers)
the method converges exponentially
PROVIDED the entries in the stiffness matrix are evaluated exactly.
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We present a new, high order accurate method for the rapid, parallel
evaluation of certain integrals in potential theory on general
three-dimensional regions. The kernels of the integrals are a
fundamental solution, or a linear combination of the derivatives
of a fundamental solution of some second order linear elliptic
differential equation. What is different and important about
these methods is that they avoid the the use of any standard
quadrature formula or the exact evaluation of any integral.
Instead, they rely on rapid methods of solving the differential
equation which the kernel satisfies. In fact, the number of operations
needed to evaluate the integrals is essentially equal to the
number of operations needed to solve the differential equation on a
uniform cubic grid.
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->[GAMM-Homepage]
The discretization of mixed boundary value problems by Galerkin or
collocation boundary element methods leads to a system of linear
equations with a dense matrix, whose spectral condition
number depends from the mesh size h.
->[Timetable]
->[GAMM-Homepage]
In the FE/BE coupling method by Schnack et al. ([1],[3]) we need
a numerical technique to map the Neumann data (traction field)
into the Dirichlet data (displacement field) on the coupling
boundary. The basis of the method is the Calderon projector
with the strong singular boundary integral equation of the
linear elasticity. To discretize the problem, we use the
collocation method. As the linear system of algebraic equations
is very sensitive to small perturbations
(e.g. roundoff error) in the matrix and in the right--hand side,
we must resort to special techniques. Türke [3] developed
an iterative algorithm to solve the linear system based on the
Neumann series. In the lecture we present the efficient
parallelized version of this iterative method. The algorithm
is well suited for MIMD parallel architectures. We examine the
convergence properties and compare some message passing communication
structures on several test problems.
[1] SCHNACK E.: "Stress Analysis with a Combination of HSM and BEM"
Proceedings of the MAFELAP 1984 Conference on "The Mathematics of Finite
Elements and Applications", Uxbridge, 1-4 May 1984. Edited by J. R.
Whiteman, Academic Press 1985, pp. 273-281.
[2] KARAOSMANOGLU N.: "Kopplung von Randelement- und Finite Element Verfahren für
dreidimensionale elastische Strukturen"
Ph.D. thesis at the Institute of Solid Mechanics,
Karlsruhe University (1989).
[3] TÜRKE K.: "Eine Zweigitter-Methode zur Kopplung von FEM und BEM"
Ph.D. thesis at the Institute of Solid Mechanics,
Karlsruhe University (1995).
->[Timetable]
->[GAMM-Homepage]
A coupling algorithm of FEM and BEM for solving mixed boundary value
problems in elas\-tostatics is presented. A domain decomposition of
the bounded domain Omega leads to a basic substructure Omega_1
with a skeleton H, where
the FEM is applied, and several macroelements
Omega_i (i=2,...,p), where the BEM is used on a fine
grid h.
As fundamental equations the well known energy bilinear
form for the whole domain Omega and, additionally, a second bilinear
form on the macroelements
Omega_i as a coupling condition is used.
This way a symmetric and nonconforming
-- that means different, independent grids H and h -- coupling
algorithm can be obtained.
This can be realized by applying the Poincaré--Steklov
operator on the macroelement surfaces in the strong singular form and by
avoiding the use of the hypersingular boundary integral equation.
[1] Kupradze V.D., T.V. Burchuladze, T.G. Gegelia, M.O. Bacheleishvilii:
Three dimensional problems of elasticity and thermoelasticity.
North--Holland, Amsterdam (Nauka, Moscow, 1976).
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The boundary integral equation for the missing Cauchy datum can
be used to compute arbitrary derivatives of the original solution
with a boot--strapping algorithm. A related algorithm leads to a
window technique which can be used for local error indicators as
well as for overlapping Schwarz methods. The lecture presents
joint work with C. Schwab and H. Schulz.
->[Timetable]
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The principal bottleneck of many 3D industrial boundary integral
applications is related to necessity to store using O(n^2) memory
locations and to solve by a direct solver using O(n^3) arithmetic
operations large dense linear systems. We suggest a so-called algebraic
matrix free iterative solution strategy with an O(n^alpha) alpha < 2,
serial arithmetic complexity which requires only O(n^beta), beta < 2,
memory locations. The main idea is to replace the original linear system
AX = B by the so-called compressed linear system tilde{A} tilde{X} = B,
where only O(n^beta), beta < 2, memory locations is required to store
tilde{A} and only O(n^alpha), alpha < 2, multiplications is required
to compute tilde{A} Y. At the same time we quarantee that
||tilde{X} - X||/||X|| < epsilon for any prescribed epsilon if
||A - tilde{A}||_F < = epsilon_A ||A||_F, where tilde{A} is
a block low rank matrix approximation. This strategy allows to solve on
existing supercomputers very large complex linear systems of sizes up to
several hundreds thousands. For example, the complex dense linear system of
size n = 80802 with 722 multiple right hand sides originated from the
standard Metallic NASA Almond CEM benchmark can be solved on 1 CPU of CRAY
C90 in 25h 20min by the matrix free iterative solver LRA_CDENSE using only
147 MWords of the main memory and 21.2 GBytes of the disk space. The incore
direct solver LAPACK will require more than 104 GBytes and more than 390h
(assuming its peak performance on CRAY C90). The real dense unsymmetric
linear system of size n = 154089 with a single right hand side originated
from the industrial CFD benchmark can be solved on 1 CPU of CRAY C90 in 7h
by the matrix free iterative solver LRA_RDENSE using only 103 Mwords of
the main memory and 13 GBytes of the disk space. The incore direct solver
LAPACK will require more than 192 Gbytes and more than 685h.
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A wavelet algorithm for the BEM corresponding to the fixed
Andreas Rathsfeld
boundary value problem of geodesy
Weierstraß-Institut für Angewandte Analysis und Stochastik,
Mohrenstr.39, Berlin 10117
\begin{eqnarray} \label{1}
-2\pi\cdot\cos [n(x),l(x)]\cdot f(x) &&
\nonumber \vspace*{5mm}
\nonumber
+ p.v.\int_\Gamma
\frac{\cos [l(x),y-x]}{|y-x|^2}\cdot f(y)\cdot d\Gamma (y)&=&4\pi\cdot
\delta g(x).
\end{eqnarray}
Fast Algorithms for Galerkin BEM:
Stefan A. Sauter,
Panel-Clustering and Cubature
Lehrstuhl Praktische Mathematik, Mathematisches Seminar
Christian-Albrechts-Universität Kiel
Hermann-Rodewald-Str. 3, D-24098 Kiel
A symmetrical boundary element formulation
H. Schippers, F.P. Grooteman
through cabin walls
National Aerospace Laboratory NLR
P.O. Box 90502, 1006 BM AMSTERDAM, The Netherlands
In this paper a symmetrical boundary element is described for the trans-
mission of sound through a panel of a cabin wall. The geometrical model
consists of a double panel configuration including a cavity partly filled
with porous material and with air. The lower panel is a stiffened skin
panel, while the upper panel is an un-stiffened trim panel. The trim panel
is backed by a semi-infinite room modelling the interior of the aircraft.
The displacement of the skin panel is assumed to be given by the exterior
acoustic field at the outer side of the aircraft. The sound transmission
problem requires the calculation of the acoustic pressure in the air and in
the porous material as well as the fluid-structure interaction with the
un-stiffened trim panel. The acoustic pressure in the air and porous
material is modelled by the Helmholtz equation (in the case of glass wool
insulation with a complex wavenumber, which models the damping). The porous
material is essentially assumed to be described by its porosity, resistivi-
ty, effective density and effective speed of sound, all of which are
frequency dependent.
The acoustic pressure in the vibrating medium (air and porous material) is
modelled by a boundary integral ansatz. The vibrating panels and the
vibrating medium are coupled by the boundary condition for flexible walls.
Classical boundary integral formulations result in a non-symmetric
boundary element matrix, while the discretization of the Helmholtz equation
using finite elements would yield a symmetrical stiffness matrix. In the
present paper a symmetrical boundary integral formulation is described for
the acoustic pressure inside and outside the cabin wall. The formulation
maps the normal derivative of the pressure along the boundary on the
pressure itself. In the case of a bounded domain (e.g. the volume covered
by the porous medium or the volume covered by the air inside the cabin
wall) the symmetrical boundary integral formulation has been taken from
$[1]$. The symmetrical boundary element matrix follows from applying
Galerkin's method. The present formulation is advantageous from the point
of view that the boundary element matrix has the same mapping properties
as the finite element matrix of the weak formulation of the Helmholtz
equation. These properties are not preserved by the classical boundary
element formulation, which yields an asymmetric system matrix. The basis
and test functions are given by the classical piecewise linear functions
on flat non-overlapping triangular elements.
The boundary elements are put on the panels of the cabin wall and on the
interface between the air and the porous material. The distance between the
panels and the interface is small, which requires an accurate evaluation
of the influence coefficients of the boundary element matrix. Cubature
formulas for the nearly singular surface integrals will be discussed.
The computations involve the discretization of a hyper-singular integral
operator, a weakly singular integral operator and a regular operator. The
hyper-singular integral operator is regularized by integration of parts.
Once this operator has been regularized, the computation of the coeffi-
cients of the corresponding boundary element matrix requires the same
number of kernel evaluations as the computation of the coefficients of the
weakly singular operator. Therefore, the computational costs of the
present symmetrical boundary element formulation are comparable with the
costs of classical boundary element formulations in acoustics, which are
based on the direct method.
Multi-Scale Methods for Boundary Integral Equations
Reinhold Schneider
Fachbereich Mathematik
Technische Hochschule Darmstadt
Schloßgartenstraße 7, Darmstadt, Germany
An outlook over essential implementation features and
particularities of wavelet basis approach
including pre-processing for surface and mesh generation, hierarchical
data structure,
a priory compression strategy, adaptive feedback
matrix computation or oracle, adaptive boundary element
methods and application to post-processing
for field computation close and far from the boundary
should be appended.
Quadrature error analysis for hp-Galerkin BEM on polyhedra
Christoph Schwab
Seminar fuer Angewandte Mathematik, ETH Zürich,
Rämistrasse 101, CH-8092 Zürich
We propose a numerical quadrature scheme for the
fully discrete evaluation of the singular
and near singular integrals in the stiffness matrix.
We pay particular attention to the numerical evaluation
of the high aspect ratio elements arising in the vicinity
of the edges of the polyhedron.
We show that using certain regularizing coordinate
transformations combined with proper geometric subdivisions
of the domains of integration all entries in the stiffness
matrix can be computed to the required exponential accuracy
with work that grows only algebraically in terms of the
number of degrees of freedom.
We present numerical computations which support
the theoretical predictions.
Rapid, Parallel Evaluation of Integrals in Potential Theory on
Ann Greenbaum, Anita Mayo, Vijay Sonnad
General Three Dimensional Regions
In particular, one can evaluate integrals whose kernels are the Green's
function for Poisson's equation by using Fourier methods on a cubic
grid, or a fast Poisson solver. Furthermore, the method requires no
evaluation of the kernel. Before applying the Poisson solver one only
needs to compute special correction terms to the right hand side of
the Poisson equation at mesh points near the boundary of the irregular
region, and these correction terms can be obtained by local calculations.
(We note that it is not possible to to evaluate these integrals by
straightforward use of Poisson solvers since they have discontinuities
in their second derivatives across the region of integration.)
The method we present can also be thought of as a way of solving elliptic
differential equations whose solutions and gradients are continuous, but
which have discontinuities in their second derivatives across some
irregular boundary.
In the talk we present results of numerical experiments, including
some on the IBM SP2 parallel computer.
Fast Solvers for Boundary Element Methods:
Olaf Steinbach
Parallelization and Preconditioning
Mathematisches Institut A
Universität Stuttgart
To construct efficient iterative solvers the use of good preconditioners
is necessary. We will give a general approach based on the
Calderon projector related to the problem. However, the construction
of fast solvers for mixed boundary value problems depends strongly on
the boundary integral formulation, which is in general not unique.
Due to the effort of integration of singular integral operators a
parallelization of these methods seems to be more suitable. Here we
consider algebraic block decompositions of the matrix as well as
domain decomposition methods. In both cases we have to discuss
related preconditioners.
Some numerical results concerning the potential equation and problems
in linear elasticity will be given.
A parallel, iterative Neumann solver for
Szabolcs Szikrai
three dimensional elastostatic problems
Institute of Solid Mechanics, University of Karlsruhe
A Two Grid Method for Coupling FEM and BEM in Elasticity
K. Türke, E.Schnack
Institute of Solid Mechanics, Karlsruhe University
The construction of a robust and reliable numerical algorithm depends on the
adaptive control of symmetry and definiteness of the coupling matrix. Therefore
we use an iterative method for solving the boundary integral equation by
expanding the {\em Calderon} projector in a Neumann series. The
proof of convergence for this expansion is based on the fundamental work
of Kupradze [1].
Numerical results in 2D and 3D will show the preciseness and efficiency
of the method.
An Extraction and Window Technique for BEM
W.L. Wendland
Universität Stuttgart
Matrix free iterative solution strategies
A. Yeremin
for 3d industrial BEM applications
Institute of Numerical Mathematics
of the Russian Academy of Sciences
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