A Monte Carlo solution method for linear elasticity

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.

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

Finite element and boundary element contact stress analysis with remeshing technique

This paper presents a symmetric coupling between the finite element and boundary element methods for analysing open microwave devices. The proposed finite element boundary element (FEBE) method is based on the E-field vector Helmholtz equation. Second-order hierarchical Curl-conforming vector finite elements over tetrahedrons are used to discretize the interior region. The boundary element (BE) truncation surface is discretized with the second-order divergence-conforming surface elements with triangular support. The symmetry of the BE part is restored through the application of the Calderon-projector. Moreover, the BE computations are accelerated using a single level QR algorithm. This reduces both memory and computational time. The resulted symmetric system of equations is solved with a very efficient preconditioned conjugate gradient (PCCG) method with a p-multiplicative Schwarz (pMUS) preconditioner. Copyright © 2005 John Wiley & Sons, Ltd.

The use of modal methods to determine acoustic transient and harmonic radiation and scattering from fluid loaded shells of revolution will be reviewed. Using such methods the velocity field of a fluid loaded shell is first expressed as an invacuo modal vector expansion where the modes are determined either analytically for simple structures or via finite element methods for more complex structures. Boundary element or other methods are subsequently used to evaluate mode-dependent acoustic radiation impedances and impulse responses which account for the coupling of the fluid. In contrast to the harmonic problem which results in sets of algebraic equations for the coefficients of the modal expansion, the transient problem results in sets of coupled time-dependent convolution integral equations for the time-dependent coefficients. Our experience in solving both two- and three-dimensional radiation and scattering problems using modal methods will be reviewed. Numerical results for a wide variety of two- and three-dimensional radiation and scattering problems will be presented. In particular results will be presented and discussed for cylindrical, spherical, and spheroidal shells.

Analysis of problems in elasto-plasticity via an adaptive FEM–BEM coupling method

The structural behavior of modern wind turbines has reached a very high complexity and many factors are involved: slenderness of the structure, excitation environment and operational controls. Moreover, if a project is located at sites with relevant seismic hazard, the wind unit must be designed considering a reasonable likelihood of earthquake occurrence during the operational state or an emergency shutdown. The influence of the subsoil on the seismic response of a wind turbine can be crucial during the seismic design phase and need to be properly included into the computational model. Norms and guidelines need to keep up with technological developments and structural peculiarities. However the dynamic soilstructure interaction is often neglected or roughly mentioned. It is usually suggested to represent the soil through springs. The proposed investigation estimates the seismic response of a soil-turbine system and involves a 1.5-MW, 3- blade wind turbine, grounded on a layered half space. The wind turbine system is modeled by means of Finite Element Method (FEM). The effects of the layering are investigated. The soil is simply idealized as a generalized spring, according to the majority of standard codes. In parallel, the same investigation is performed with a more accurate method, a coupling between finite element and Boundary Element Method (BEM). This allows assessing the applicability and accuracy of the simplified soil representation.

A coupled system of finite elements and boundary elements is presented for the modelling of sound propagation in a nonuniform mean flow. Finite elements are used to discretize the acoustic pressure in a bounded computational domain including the nonuniform flow field. The radiation of acoustic waves in the exterior uniform flow field is modelled by a boundary integral representation formula. The interface condition on the shared boundary yields an integral equation which is solved using boundary elements. The coupling of finite elements and boundary elements is described in detail for sound propagation in a stationary fluid where the acoustic pressure is governed by the Helmholtz equation. The coupled computational system is applied to estimate scattering and refractive effects on the acoustic pressure waves through a boundary layer surrounding an aircraft’s fuselage. The coupled mathematical formulation for three dimensional sound radiation around an engine in subsonic flow is discussed.

This paper analyzes the rate of convergence of the h-p version of the coupling of the finite element and boundary element method for transmission problems with a linear differential operator with variable coefficients in a bounded polyhedral domain \(\Omega_1\) and with constant coefficients in the exterior domain \(\Omega_2 = {\Bbb R}^3 \backslash \overline{\Omega}_1\). This procedure uses the variational formulation of the differential equation in \(\Omega_1\) and involves integral operators on the interface between \(\Omega_1\) and \(\Omega_2\). The finite elements are used to obtain approximate solutions of the differential equation in \(\Omega_1\) and the boundary elements are used to obtain approximate solutions of the integral equations. For given piecewise analytic data we show that the Galerkin solution of this coupling procedure converges exponentially fast in the energy norm if the h-p version is used both for finite elements and boundary elements.

A new simulation method is presented to improve the accuracy of 3-D electron trajectory calculations with the space charge effect, using both the finite element method and boundary element method to calculate the electric field and the magnetic field. In this method, the magnetic potential is first calculated by the finite element method, then calculated again by the boundary element method, and finally the magnetic field is calculated at the nodes and interpolated in the finite elements. Next, the electric field is calculated in the same manner used for the magnetic field. This method was specifically designed for the electron focusing system of an electron beam apparatus.

In the present work, graded finite element and boundary element methods capable of modeling behaviors of structures made of nonhomogeneous functionally graded materials (FGMs) composed of two constituent phases are presented. A numerical implementation of Somigliana’s identity in two-dimensional displacement fields of the isotropic nonhomogeneous problems is presented using the graded elements. Based on the constitutive and governing equations and the weighted residual technique, effective boundary element formulations are implemented for elastic nonhomogeneous isotropic solid models. Results of the finite element method are derived based on a Rayleigh–Ritz energy formulation. The heterogeneous structures are made of combined ceramic–metal materials, in which the material properties vary continuously along the in-plane or thickness directions according to a power law. To verify the present work, three numerical examples are provided in the paper.

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.

In order to assess the discretization error of a finite element solution, asymptotic solutions for predicted natural frequencies of two-dimensional elastic solid vibration problems in the finite element analysis are presented in this paper. Since the asymptotic solution is more accurate than the original finite element solution, it can be viewed as an alternative solution against which the original finite element solution can be compared. Consequently, the discretization error of the finite element solution can be evaluated. Due to the existence of two kinds of two-dimensional problems in engineering practice, both the plane stress problem and the plane strain problem have been considered and the corresponding asymptotic formulae for predicted natural frequencies of two-dimensional solids by the finite element method have been derived from the fact that a discretized finite element system approaches a continuous one if the finite element size approaches zero. It has been demonstrated, from the related numerical results of three examples, that the present asymptotic solution, which can be obtained by simply using the corresponding formula without any further finite element calculation, is indeed more accurate than the original finite element solution so that it can be considered as a kind of corrected solution for the discretization error estimation of a finite element solution.

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.

In this paper, a coupled model based on finite element method (FEM), boundary element method (BEM) and scaled boundary FEM (SBFEM) (also referred to as the consistent infinitesimal finite element cell method) for dynamic response of 2D structures resting on layered soil media is presented. The SBFEM proposed by Wolf and Song (Finite-element Modelling of Unbounded Media. Wiley: England, 1996) and BEM are used for modelling the dynamic response of the unbounded media (far-field). The standard FEM is used for modelling the finite region (near-field) and the structure. In SBFEM, which is a semi-analytical technique, the radiation condition at infinity is satisfied exactly without requiring the fundamental solution. This method, also eliminates the need for the discretization of interfaces between different layers. In both SBFEM and BEM, the spatial dimension is decreased by one. The objective of the development of this coupled model is to combine advantages of above-mentioned three numerical models to solve various soil-structure interaction (SSI) problems efficiently and effectively. These three methods are coupled (FE-BE-SBFEM) via substructuring method, and a computer programme is developed for the harmonic analyses of SSI systems. The coupled model is established in such a way that, depending upon the problem and far-field properties, one can choose BEM and/or SBFEM in modelling related far-field region(s). Thus, BEM and/or SBFEM can be used efficiently in modelling the far-field. The proposed model is applied to investigate dynamic response of rigid and elastic structures resting on layered soil media. To assess the proposed SSI model, several problems existing in the literature are chosen and analysed. The results of the proposed model agree with the results presented in the literature for the chosen problems. The advantages of the model are demonstrated through these comparisons.