scholarly journals Diagonal scaling of stiffness matrices in the Galerkin boundary element method

2000 ◽  
Vol 42 (1) ◽  
pp. 141-150 ◽  
Author(s):  
Mark Ainsworth ◽  
Bill McLean ◽  
Thanh Tran
Keyword(s):  
Integral Equation ◽  
Stiffness Matrix ◽  
Numerical Experiments ◽  
Degrees Of Freedom ◽  
Boundary Integral ◽  
Piecewise Constant ◽  
Stiffness Matrices ◽  
Diagonal Scaling ◽  
Mesh Ratio ◽  
Galerkin Boundary Element Method

AbstractA boundary integral equation of the first kind is discretised using Galerkin's method with piecewise-constant trial functions. We show how the condition number of the stiffness matrix depends on the number of degrees of freedom and on the global mesh ratio. We also show that diagonal scaling eliminates the latter dependence. Numerical experiments confirm the theory, and demonstrate that in practical computations involving strong local mesh refinement, diagonal scaling dramatically improves the conditioning of the Galerkin equations.

A Strategy for Fine Mesh Resolution in Contact Mechanics

2021 ◽  
Author(s):  
Gaurav Chauda ◽  
Daniel J. Segalman
Keyword(s):  
Degrees Of Freedom ◽  
Contact Problems ◽  
Boundary Integral ◽  
System Of Equations ◽  
Element Mesh ◽  
Contact Patch ◽  
Linear System Of Equations ◽  
Fine Mesh ◽  
Stiffness Matrices ◽  
Analytic Expressions

Abstract To obtain detail in elastic, frictional contact problems involving contact many — at least tens, and more suitably hundreds [1] — of nodes are necessary over the contact patch. Generally, this fine discretization results in intractable numbers of system equations that must be solved, but this problem is greatly mitigated when the elasticity of the contacting bodies is represented by elastic compliance matrices rather than stiffness matrices. An examination of the classical analytic expressions for the contact of disks — an example of smooth contact — shows that for most standard engineering metals, such as brass, steel, or titanium, the pressures that would cause more than one degree of arc of contact would push the materials past the elastic limit. The discretization necessary to capture the interface tractions would be on the order of at least tens of nodes. With the resulting boundary integral formulation would involve several hundreds of nodes over the disk, and the corresponding finite element mesh would have tens of thousands. The resulting linear system of equations must be solved at each load step and the numerical problem becomes extremely difficult or intractable. A compliance method of facilitating extremely fine contact patch resolution can be achieved by exploiting Fourier analysis and the Michell solution. The advantages of this compliance method are that only degrees of freedom on the surface are introduced and those not in the region of contact are eliminated from the system of equations to be solved.

scholarly journals Exact and Direct Modeling Technique for Rotor-Bearing Systems with Arbitrary Selected Degrees-of-Freedom

1994 ◽  
Vol 1 (6) ◽  
pp. 497-506 ◽  
Author(s):  
Shilin Chen ◽  
Michel Géradin
Keyword(s):  
Stiffness Matrix ◽  
Degrees Of Freedom ◽  
Dynamic Stiffness ◽  
Modeling Technique ◽  
Dynamic Stiffness Matrix ◽  
Global Dynamic ◽  
Combination Methods ◽  
Stiffness Matrices ◽  
Direct Modeling ◽  
Rotor Bearing

An exact and direct modeling technique is proposed for modeling of rotor-bearing systems with arbitrary selected degrees-of-freedom. This technique is based on the combination of the transfer and dynamic stiffness matrices. The technique differs from the usual combination methods in that the global dynamic stiffness matrix for the system or the subsystem is obtained directly by rearranging the corresponding global transfer matrix. Therefore, the dimension of the global dynamic stiffness matrix is independent of the number of the elements or the substructures. In order to show the simplicity and efficiency of the method, two numerical examples are given.

AN INNOVATIVE BOUNDARY ELEMENT FORMULATION FOR BUILDING SLABS

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Ahmed U. Abdelhady ◽  
Youssef F. Rashed
Keyword(s):  
Boundary Element ◽  
Stiffness Matrix ◽  
Degrees Of Freedom ◽  
Research Work ◽  
Element Formulation ◽  
Supporting Cells ◽  
Stiffness Analysis ◽  
Stiffness Matrices ◽  
Limited Applicability ◽  
Generalized Displacements

Slab supported by beams (i.e. beam-slab floor) is a common practice in the construction of buildings. Modeling this slab type using the boundary element method (BEM) is an essential step to provide seamless frameworks for the analysis of buildings with complex geometries. However, one of the most challenging difficulties that have been facing research efforts in this area is its theoretical setting with limited applicability to practical building slabs. This limitation is addressed in this research work and a practical BEM-based formulation for the slab-beam floor is presented. The presented formulation discretizes the connection area between the slab, and beams and columns into cells (supporting cells). The centroids of these cells are used to carry out an additional collocation scheme which is required to solve the resulting system of equations. To generate the slab stiffness matrix, the generalized displacements that correspond to the supporting cells’ degrees of freedom are set to unity (one at a time). By assembling the generated slab stiffness matrix with beams and columns stiffness matrices using the stiffness analysis method, the overall stiffness matrix is obtained. Hooke’s law is then applied to calculate the generalized displacements and straining actions are obtained in the post-processing phase. The developed formulation is applied to an example to validate its results by comparing it with the analytical solution.

Finite Element Analysis of Coupled Lateral and Torsional Vibrations of a Rotor With Multiple Cracks

2005 ◽  
Author(s):  
Xi Wu ◽  
Jerzy T. Sawicki ◽  
Michael I. Friswell ◽  
George Y. Baaklini
Keyword(s):  
Stiffness Matrix ◽  
Degrees Of Freedom ◽  
Accurate Determination ◽  
Crack Model ◽  
Multiple Cracks ◽  
Torsional Vibrations ◽  
Element Analysis ◽  
Cosine Series ◽  
Six Degrees Of Freedom ◽  
Stiffness Matrices

The coupling between lateral and torsional vibrations has been investigated for a rotor dynamic system with breathing crack model. The stiffness matrix has been developed for the shaft element which accounts for the effect of the crack and all six degrees of freedom per node. Since the off-diagonal terms of the stiffness matrix represent the coupling of the respective modes, the special attention has been paid on accurate determination of their values. Based on the concepts of fracture mechanics, the variation of the stiffness matrix over the full shaft revolution is represented by the truncated cosine series where the fitting coefficient matrices are extracted from the stiffness matrices of the cracked shaft for a number of its different angular positions. The variation of the system eigenfrequencies and dynamic response of the rotor with two cracks have been studied for various shaft geometries, crack axial locations, and relative phase of cracks.

scholarly journals APPLICATION OF COLLOCATION BEM FOR AXISYMMETRIC TRANSMISSION PROBLEMS IN ELECTRO- AND MAGNETOSTATICS

2016 ◽  
Vol 21 (1) ◽  
pp. 16-34 ◽  
Author(s):  
Olga Lavrova ◽  
Viktor Polevikov
Keyword(s):  
Integral Equation ◽  
Boundary Integral Equations ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Meridian Plane ◽  
Piecewise Constant ◽  
Boundary Interpolation ◽  
Transmission Problems ◽  
Polygonal Boundary

This paper considers the numerical solution of boundary integral equations for an exterior transmission problem in a three-dimensional axisymmetric domain. The resulting potential problem is formulated in a meridian plane as the second kind integral equation for a boundary potential and the first kind integral equation for a boundary flux. The numerical method is an axisymmetric collocation with equal order approximations of the boundary unknowns on a polygonal boundary. The complete elliptic integrals of the kernels are approximated by polynomials. An asymptotic kernels behavior is analyzed for accurate numerical evaluation of integrals. A piecewise-constant midpoint collocation and a piecewise-linear nodal collocation on a circular arc and on its polygonal interpolation are used for test computations on uniform meshes. We analyze empirically the influence of the polygonal boundary interpolation to the accuracy and the convergence of the presented method. We have found that the polygonal boundary interpolation does not change the convergence behavior on the smooth boundary for the piecewise-constant and the piecewise-linear collocation.

scholarly journals Piecewise constant collocation for first-kind boundary integral equations

1994 ◽  
Vol 35 (3) ◽  
pp. 396-398
Author(s):  
I. G. Graham ◽  
Y. Yan
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Parameter Space ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Piecewise Constant ◽  
Break Points ◽  
Image Position ◽  
A Minor ◽  
Closed Polygon

We wish to correct a minor error in the recent paper [2]. That paper was concerned with an integral equation defined on a closed polygon Γ with r corners at the points x0, x2, …, x2r = x0. We parameterized Γ using a mapping γ:[−π,π] → Γ defined as follows. For each l, introduce the mid-point x2l−1 of the side joining x2l—2 to x2l. Then introduce 2r + 1 points in parameter spacewith the property that for each j = 1, …, 2rwhere mj are integers and . Then γ(s) is defined byfor j = 1, …, 2r. The {Sj} are then the preimages of the {xj} under γ. Moreover, in view of (1), a family of uniform meshes can be constructed on [−π, π] which include {Sj} as the break-points. Then γ maps these to meshes which are uniform on each segment joining xj−1 to xj (which we denote Γj). These meshes are used to discretize the integral equation.

scholarly journals An offline-online strategy for multiscale problems with random defects

2022 ◽  
Author(s):  
Axel Målqvist ◽  
Barbara Verfürth
Keyword(s):  
Stiffness Matrix ◽  
Numerical Experiments ◽  
Linear Combinations ◽  
Reference Element ◽  
Multiscale Problems ◽  
Stiffness Matrices ◽  
Online Strategy ◽  
Local Defects ◽  
Random Defects ◽  
Selection Of

In this paper, we propose an offline-online strategy based on the Localized Orthogonal Decomposition (LOD) method for elliptic multiscale problems with randomly perturbed diffusion coefficient. We consider a periodic deterministic coefficient with local defects that occur with probability $p$. The offline phase pre-computes entries to global LOD stiffness matrices on a single reference element (exploiting the periodicity) for a selection of defect configurations. Given a sample of the perturbed diffusion the corresponding LOD stiffness matrix is then computed by taking linear combinations of the pre-computed entries, in the online phase. Our computable error estimates show that this yields a good approximation of the solution for small $p$, which is illustrated by extensive numerical experiments.  This makes the proposed technique attractive already for moderate sample sizes in a Monte Carlo simulation.

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