John P. Wolf

The scaled boundary finite-element method—alias consistent infinitesimal finite-element cell method—for elastodynamics

The scaled boundary finite-element method, alias the consistent infinitesimal finite-element cell method, is developed starting from the governing equations of linear elastodynamics. Only the boundary of the medium is discretized with surface finite elements yielding a reduction of the spatial dimension by one. No fundamental solution is necessary, and thus no singular integrals must be evaluated. General anisotropic material is analysed without any increase in computational effort. Boundary conditions on free and fixed surfaces and on interfaces between different materials are enforced exactly without any discretization. This method is exact in the radial direction and converges to the exact solution in the finite-element sense in the circumferential directions. For a bounded medium symmetric static-stiffness and mass matrices with respect to the degrees of freedom on the boundary result without any additional assumption. A stress singularity is represented very accurately, as the condition on the boundary in the vicinity of the point of singularity is satisfied without spatial discretization.

The scaled boundary finite-element method – a primer: derivations

Abstract The scaled boundary finite-element method is a semi-analytical fundamental-solution-less boundary-element method based solely on finite elements. Using the simplest wave propagation problem and discretizing the boundary with a two-node line finite element, which preserves all essential features, two derivations of the scaled boundary finite-element equations in displacement and dynamic stiffness are presented. In the first, the scaled-boundary-transformation-based derivation, the new local coordinate system consists of the distance measured from the so-called scaling centre and the circumferential directions defined on the surface finite element. The governing partial differential equations are transformed to ordinary differential equations by applying the weighted-residual technique. The boundary conditions are conveniently formulated in the local coordinates. In the second, the mechanically based derivation, a similar fictitious boundary is introduced. A finite-element cell is constructed between the two boundaries. Standard finite-element assemblage and similarity lead to the scaled boundary finite-element equations after performing the limit of the cell width towards zero analytically.

Semi-analytical representation of stress singularities as occurring in cracks in anisotropic multi-materials with the scaled boundary finite-element method

The scaled boundary finite-element method, a semi-analytical boundary-element method based on finite elements, is applied to fracture mechanics problems. Only the actual boundary of the body, but not the straight crack faces and material interfaces passing through the crack tip, is spatially discretized with finite elements, leading to a reduction of the spatial dimension by one. In the radial direction the displacements follow from the analytical solution without any a priori assumption. The boundary condition on the crack faces and material interfaces are satisfied exactly. In this boundary-element method based on finite elements, no fundamental solution is required, which permits an efficient analysis of anisotropic material and avoids singular integrals. The scaled boundary finite-element method thus permits an accurate representation of singularities in the radial direction using analytical functions.

Spring-dashpot-mass models for foundation vibrations

Note: [228a] Reference LCH-ARTICLE-1997-012doi:10.1002/(SICI)1096-9845(199709)26:9 3.0.CO;2-MView record in Web of Science Record created on 2007-04-24, modified on 2016-08-08

CONSISTENT INFINITESIMAL FINITE‐ELEMENT CELL METHOD: THREE‐DIMENSIONAL VECTOR WAVE EQUATION

To calculate the unit-impulse response matrix of an unbounded medium for use in a time-domain analysis of unbounded medium–structure interaction, the consistent infinitesimal finite-element cell method is developed for the three-dimensional vector wave equation. This is a boundary finite-element procedure. The discretization is only performed on the structure–medium interface, yielding a reduction of the spatial dimension by 1. The procedure is rigorous in the radial direction and exact in the finite-element sense in the circumferential directions. In contrast to the boundary-element procedure, the consistent infinitesimal finite-element cell method does not require a fundamental solution and incorporates interfaces extending from the structure–medium interface to infinity compatible with similarity without any additional computational effort. A general anisotropic material can be processed. The derivation is based on the finite-element formulation and on similarity.

The scaled boundary finite-element method – a primer: solution procedures

Abstract The scaled boundary finite-element equations in displacement and dynamic stiffness, which are ordinary differential equations, derived in the accompanying paper involve the discretization of the boundary only. The general solution procedure is demonstrated addressing an illustrative example which consists of a two-dimensional out-of-plane (anti-plane) motion with a single degree of freedom on the boundary. For statics and dynamics in the frequency domain, the displacements in the domain and the stiffness matrix with degrees of freedom on the boundary only are obtained analytically for bounded and unbounded media. The radiation condition is satisfied exactly using the high-frequency asymptotic expansion for the dynamic-stiffness matrix of an unbounded medium. The mass matrix for a bounded medium is determined analytically. Body loads in statics are calculated analytically. Numerical procedures to calculate the dynamic-stiffness and unit-impulse response matrices for an unbounded medium are also presented. The scaled boundary finite-element method is semi-analytical as the ordinary differential equations in displacement are solved analytically, which permits an efficient calculation of displacements, stresses and stress intensity factors. This boundary-element method based on finite elements leads to a reduction of the spatial dimension by one. As no fundamental solution is required, no singular integrals are evaluated and anisotropic material is analysed without additional computational effort.

The scaled boundary finite-element method: analytical solution in frequency domain

The scaled boundary finite-element equation, a system of ordinary differential equations in the radial coordinate, is solved analytically applying a series expansion. An eigenvalue problem of a Hamiltonian matrix is solved. When the difference of two eigenvalues is an even number, besides the power series, logarithmic functions occur. Two sets of independent solutions are obtained. The scaled boundary finite-element method is thus a semi-analytical procedure to solve partial differential equations with the polynomial approximation of finite elements in the two circumferential directions and a closed-form solution in the radial direction. Applications to bounded and unbounded domains demonstrate excellent accuracy.

The scaled boundary finite-element method – a fundamental solution-less boundary-element method☆

In this boundary-element method based on finite elements only the boundary is discretized with surface finite elements yielding a reduction of the spatial dimension by one. No fundamental solution is necessary and thus no singular integrals must be evaluated and general anisotropic material can be analysed. For an unbounded (semi-infinite or infinite) medium the radiation condition at infinity is satisfied exactly. No discretization of free and fixed boundaries and interfaces between different materials is required. The semi-analytical solution inside the domain leads to an efficient procedure to calculate the stress intensity factors accurately without any discretization in the vicinity of the crack tip. Body loads are included without discretization of the domain. Thus, the scaled boundary finite-element method not only combines the advantages of the finite-element and boundary-element methods but also presents appealing features of its own. After discretizing the boundary with finite elements the governing partial differential equations of linear elastodynamics are transformed to the scaled boundary finite-element equation in displacement, a system of linear second-order ordinary differential equations with the radial coordinate as independent variable, which can be solved analytically. Introducing the definition of the dynamic stiffness, a system of nonlinear first-order ordinary differential equations in dynamic stiffness with the frequency as independent variable is obtained. Besides the displacements in the interior the static-stiffness and mass matrices of a bounded medium and the dynamic-stiffness and unit-impulse response matrices of an unbounded medium are calculated.

Some cornerstones of dynamic soil-structure interaction

Abstract Salient features of dynamic soil–structure interaction are discussed. A criterion for the presence of radiation damping in a site is formulated. The radiation condition at infinity of outwardly propagating energy can for certain sites correspond to incoming waves. The consequences that the dynamic behaviour of the unbounded soil depends on the dimensionless frequency, which is proportional to the product of the frequency and the radial coordinate, are discussed. The partition of the radiated energy of surface waves and of body waves for increasing frequency is addressed. In addition, procedures to analyse the dynamic soil–structure interaction are outlined, ranging from the approximate simple physical models (cones, spring–dashpot–mass representations) for the soil to the damping-solvent extraction method and to the rigorous forecasting method and the scaled boundary finite-element method. Convolution integrals can be avoided by constructing a dynamic system with a finite number of degrees of freedom for the soil. Extensions for moving concentrated loads and an increase in efficiency using a reduced set of base functions are presented. The damping ratio of an equivalent one-degree-of-freedom system representing the interaction of the structure with the soil for a horizontal earthquake reflects the effect of the cutoff frequency for a soil layer. The same simple model can be extended to consider the partial uplift of a basemat for seismic excitation (nonlinear soil–structure interaction). Two-dimensional versus three-dimensional foundation modelling is examined.

Body loads in scaled boundary finite-element method

The solution of the scaled boundary finite-element equation in displacement with body loads is derived. The non-homogeneous term caused by the body loads is processed using the technique of variation of parameters. Integrals in the radial direction arise which can, however, be evaluated explicitly for concentrated loads and loads varying as power functions in the radial coordinate. In these cases no additional approximations are introduced. The scaled boundary finite-element method thus remains a semi-analytical fundamental-solution-less boundary-element method based on finite elements.