Arbitrary Body Force Research Articles

Addressing arbitrary body forces in 2D elasticity coupling the Radial Basis Integration Method with boundary elements

A boundary-domain integral formulation inevitably arises when the boundary element method (BEM) is applied for solving the differential equation that governs linear elastostatic problems with body forces. Although the domain integrals introduced by the body forces can be evaluated using internal cells this destroys the boundary-only meshing feature of BEM and makes the integration processes inefficient. This paper shows that these problems can be solved more efficiently without using internal cells by the Radial Basis Integration Method (RBIM) which employs a meshless quadrature obtained by performing boundary-only offline computations. Using RBIM, weakly singular domain integrals can be computed via a simple selective quadrature procedure, whereas, strong singular domain integrals may be computed using two schemes: The first, is based on the same selective quadrature procedure, and the second is based on the singularity separation scheme. The present method, unlike classical radial integration method (RIM), does not have problems in calculating integrals in concave or multiple-connected domains. The results obtained in some 2D linear elastostatic problems with arbitrary body forces show that this method can be as accurate as RIM but less time-consuming than the latter. This method could be applied to other engineering problems involving source terms.

Analytical solutions of peristatics and peridynamics for 3D isotropic materials

In a recent paper, Mikata (2019) has established two explicit analytical governing equations of peridynamics for 3D isotropic and orthotropic materials. In the present paper, general analytical solutions of peristatics and peridynamics for 3D isotropic peridynamic materials are obtained in infinite media for an arbitrary body force density, and arbitrary initial conditions in the case of peridynamics. In addition, peristatic and peridynamic Green's functions corresponding to a point load are also obtained. Classical limit of the peristatic and peridynamic Green's functions in the limit of the generalized material horizon δ going to 0 (δ→0) is analytically discussed in detail for one of the most commonly used nonlocality functions with a finite material horizon with a sharp cut-off. It is analytically and explicitly shown that the peristatic and peridynamic Green's functions converge to the elastostatic and elastodynamic Green's functions, respectively, as the generalized material horizon δ goes to 0 (δ→0). Numerical results are obtained and discussed for the peristatic Green's function and a 3D peristatic displacement field under a 1D-like loading. The governing equation investigated in the present paper is the first explicit analytical governing equation of peridynamics for 3D isotropic materials, which is comparable to Navier-Cauchy's equation. Thus, this equation opens up various ways to investigate peridynamics in 3D isotropic materials in an exact analytical fashion, which was not possible before.

Fundamental and Generalized Solutions of the Equations of Motion of a Thermoelastic Half-Plane with a Free Boundary

A coupled thermoelasticity model is used to study the dynamics of a thermoelastic half-plane influenced by nonstationary body forces and heat sources. In space of Laplace transforms with respect to time, Green’s tensor of the boundary value problem for the half-plane with a boundary free of stresses and heat fluxes is constructed. The displacements and the temperature of the medium are determined for arbitrary body forces and heat sources.

Convection-driven kinematic dynamos with a self-consistent shear flow

ABSTRACTIt is widely accepted that astrophysical magnetic fields are generated by dynamo action. In many cases, these fields exhibit organisation on a scale larger than that of the underlying turbulent flow (e.g. the 11-year solar cycle). The mechanism for the generation of so-called large-scale fields remains an open problem. In cases where the magnetic Reynolds number (Rm) is small, dynamo-generated fields are coherent but at (the astrophysically relevant) high Rm, the fields are overwhelmed by small-scale fluctuating field. Recently Tobias and Cattaneo have shown that an imposed large-scale shear flow can suppress the small-scale fluctuations and allow the large-scale temporal behaviour to emerge. Shear is also believed to modify the electromotive force by introducing correlations between the flow and the field. However, in previous models at high Rm the shear is often artificially imposed or driven by an arbitrary body force. Here we consider a simple kinematic model of a convective dynamo in which shear is self-consistently driven by the presence of a horizontal temperature gradient (resulting in a thermal wind) and a rotation vector that is oblique to gravity. By considering a -dimensional system, we are able to reach high Rm so that the dynamo approaches the asymptotic regime where the growth rate becomes approximately independent of Rm. We find the flows studied here to be excellent small-scale dynamos, but with very little systematic behaviour evident at large Rm. We attribute this to being unable to self-consistently generate flows with both large (net) helicity and strong shear in this setup.

The Boundary Element Method with a Fast Multipole Accelerated Integration Technique for 3D Elastostatic Problems with Arbitrary Body Forces

A line integration boundary element method (LIBEM) is proposed for three-dimensional elastostatic problems with body forces. The method is a boundary-only discretization method like the traditional boundary element method (BEM), and the boundary elements created in BEM can be used directly in the proposed method for constructing the integral lines. Finally, the body forces are computed by summing one-dimensional integrals on straight lines. Background cells can be used to cut the lines into sub-lines to compute the integrals more easily and efficiently. To further reduce the computational time of LIBEM, the fast multipole method is applied to accelerate the method for large-scale computations and the details of the fast multipole line integration method for 3D elastostatic problems are given. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

A Clifford Algebra Formulation of Navier-cauchy Equation

The three-dimensional (3D) displacement field in elastic body subjected to body force usually governed by the Navier-Cauchy equation is here formulated in the language of Clifford valued functions and the Dirac operator. Using Clifford analysis and considering arbitrary body force, we solve the problem, expressing the displacement field in terms of one harmonic function and one monogenic function.

A novel 3D BE formulation for general multi-zone domains under body force loading

The current paper proposes a boundary element formulation, applicable to 2-D and 3-D elastostatics problems using a unified approach for transformations of the domain integrals into boundary integrals. The method is applicable to linear problems encompassing both finite and infinite multi-region domains allowing non-vanishing body forces. Numerical results agree quite well with the analytical solutions; while the present method offers easy formulation with less numerical efforts in comparison to FEM or some BEM which need interior points to treat arbitrary body forces. It is demonstrated that the method has the potential to have profound impact on engineering design, notably in dam-foundation interaction.

Multiquadric and Chebyshev approximation to three-dimensional thermoelasticity with arbitrary body forces

Abstract In this paper, analytical particular solutions of multiquadrics and Chebyshev polynomials associated with problems of three-dimensional thermoelasticity are derived, which provides a supplement to the review article of Cheng, Chen, Golberg and Rashed (Engineering Analysis with Boundary Elements 25 (2001), 377). In the derivation, the three coupled second-order partial differential equations (PDEs) are converted into a biharmonic equation. Then, the multiquadric and polynomial particular solutions of the biharmonic equation are obtained respectively by straight integrations and referring to the first author‘s recent study. For the multiquadric particular solutions, they are set to be infinitely differentiable by suitably arranging the coefficients of its Laurent series. And for the polynomial particular solutions, they can be represented explicitly and implemented without any book keeping. Numerical experiments are carried out to validate the derived particular solutions.

Successive elimination of shear layers by a hierarchy of constraints in inviscid spherical-shell flows

In a rotating spherical shell, an inviscid inertia-free flow driven by an arbitrary body force will have cylindrical components that are either discontinuous across, or singular on, the tangent cylinder, the cylinder tangent to the inner core and parallel to the rotation axis. We investigate this problem analytically, and show that there is an infinite hierarchy of constraints on this body force which, if satisfied, sequentially remove discontinuities or singularities in flow derivatives of progressively higher order. By splitting the solution into its equatorial symmetry classes, we are able to provide analytic expressions for the constraints and demonstrate certain inter-relations between them. We show numerically that viscosity smoothes any singularity in the azimuthal flow component into a shear layer, comprising inner and outer layers, either side of the tangent cylinder, of width O(E2/7) and O(E1/4), respectively, where E is the Ekman number. The shear appears to scale as O(E−1/3) in the equatorially symmetric case, although in a more complex fashion when considering equatorial antisymmetry, and attains a maximum value in either the inner or outer sublayers depending on equatorial symmetry. In the low-viscosity magnetohydrodynamic system of the Earth's core, magnetic tension within the fluid resists discontinuities in the flow and may dynamically adjust the body force in order that a moderate number of the constraints are satisfied. We speculate that it is violations of these constraints that excites torsional oscillations, magnetohydrodynamic waves that are observed to emanate from the tangent cylinder.

Linear and nonlinear stability of floating viscous sheets

AbstractWe study the stability of a thin, Newtonian viscous sheet floating on a bath of denser fluid. We first derive a general set of equations governing the evolution of a nearly flat sheet, accounting for geometrical nonlinearities associated with moderate rotations. We extend two classical models by considering arbitrary external body and surface forces; these two models follow from different scaling assumptions, and are derived in a unified way. The equations capture two modes of deformation, namely viscous bending and stretching, and describe the evolution of thickness, mid-surface and in-plane velocity as functions of two-dimensional coordinates. These general equations are applied to a floating viscous sheet, considering gravity, buoyancy and surface tension. We investigate the stability of the flat configuration when subjected to arbitrary in-plane strain. Two unstable modes can be found in the presence of compression. The first one combines undulations of the centre-surface and modulations of the thickness, with a wavevector perpendicular to the direction of maximum applied compression. The second one is a buckling mode; it is purely undulatory and has a wavevector along the direction of maximum compression. A nonlinear analysis yields the long-time evolution of the undulatory mode.

The method of fundamental solutions with dual reciprocity for three‐dimensional thermoelasticity under arbitrary body forces

PurposeThe purpose of this paper is to develop a meshless numerical method for three‐dimensional isotropic thermoelastic problems with arbitrary body forces.Design/methodology/approachThis paper combines the method of fundamental solutions (MFS) and the dual reciprocity method (DRM) as a meshless numerical method (MFS‐DRM) to solve three‐dimensional isotropic thermoelastic problems with arbitrary body forces. In the DRM, the arbitrarily distributed temperature and body force are approximated by polyharmonic splines with augmented polynomial basis, whose particular solutions and the corresponding tractions are reviewed and given explicitly. The MFS is then applied to solve the complementary solution. Numerical experiments of Dirchlet, Robin, and peanut‐shaped‐domain problems are carried out to validate the method.FindingsIn literature, it is commented that the Gaussian elimination can be used reliably to solve the MFS equations for non‐noisy boundary conditions. For noisy boundary conditions, the truncated singular value decomposition (TSVD) is more accurate than the Gaussian elimination. In this paper, it was found that the particular solutions obtained by the DRM act like noises and the use of TSVD improves the accuracy.Originality/valueIt is the first time that the MFS‐DRM is derived to solve three‐dimensional isotropic thermoelastic problems with arbitrary body forces.

On the lower bound of net driving power in controlled duct flows

We examine mathematically the lower bound of the net driving power (i.e., the summation of pumping and actuation powers) of a controlled duct flow under a constant flow rate. The net power in a duct with arbitrary cross-section in the presence of the inertial term, blowing/suction from the wall, and arbitrary body forces can be decomposed into four terms: (1) dissipation due to the velocity profile of the Stokes flow (in other words, pumping power for the Stokes flow); (2) dissipation due to deviation of the mean velocity from the Stokes flow profile; (3) dissipation due to velocity fluctuations; and (4) correlation between the wall-pressure of the Stokes flow and the time-averaged blowing/suction velocity. Of these, the first three terms are shown to be non-negative, while the sign of the fourth term is indefinite. The fourth term vanishes in cases where the duct has a constant-shape cross-section, such as cylindrical pipes and plane channels. Namely, in such cases, the lower bound of net power is exactly given by the dissipation rate (pumping power) of the Stokes flow at the same flow rate.

Computing stationary free-surface shapes in microfluidics

A finite-element algorithm for computing free-surface flows driven by arbitrary body forces is presented. The algorithm is primarily designed for the microfluidic parameter range where (i) the Reynolds number is small and (ii) force-driven pressure and flow fields compete with the surface tension for the shape of a stationary free surface. The free surface shape is represented by the boundaries of finite elements that move according to the stress applied by the adjacent fluid. Additionally, the surface tends to minimize its free energy and by that adapts its curvature to balance the normal stress at the surface. The numerical approach consists of the iteration of two alternating steps: The solution of a fluidic problem in a prescribed domain with slip boundary conditions at the free surface and a consecutive update of the domain driven by the previously determined pressure and velocity fields. For a Stokes problem the first step is linear, whereas the second step involves the nonlinear free-surface boundary condition. This algorithm is justified both by physical and mathematical arguments. It is tested in two dimensions for two cases that can be solved analytically. The magnitude of the errors is discussed in dependence on the approximation order of the finite elements and on a step-width parameter of the algorithm. Moreover, the algorithm is shown to be robust in the sense that convergence is reached also from initial forms that strongly deviate from the final shape. The presented algorithm does not require a remeshing of the used grid at the boundary. This advantage is achieved by a built-in mechanism that causes a smooth change from the behavior of a free surface to that of a rubber blanket if the boundary mesh becomes irregular. As a side effect, the element sides building up the free surface in two dimensions all approach equal lengths. The presented variational derivation of the boundary condition corroborates the numerical finding that a second-order approximation of the velocity also necessitates a second-order approximation for the free surface discretization.

Introduction to the Classic Paper: Body Force Equivalents for Seismic Dislocations, by R. Burridge and L. Knopoff

The Burridge and Knopoff (1964) paper played a critical role in the development of the theory of the earthquake source, and its importance is best appreciated when it is considered within that context. For this reason, a brief historical overview will be presented. A more detailed presentation and a number of basic results needed to understand the paper can be found in the accompanying tutorial (Pujol, 2003; see this issue, pp. 163-153). Although solving the elastic wave equation for an arbitrary body force is a difficult problem, solutions for single and double couples had been known since the early 1900's. The problem that seismologists …

A BEM formulation for axisymmetric elasticity with arbitrary body force using particular integrals

A particular integral formulation is presented for purely axisymmetric elasticity with arbitrary body force. The axisymmetric elastostatic equation is used for the complementary solution. The particular integrals for displacement and traction are derived by integrating three-dimensional ones along the circumferential direction. The numerical results for four example problems are given and compared with their analytical solutions to demonstrate the accuracy of the present formulation.

Three-Dimensional Stress Analysis with Arbitrary Body Force by Triple-Reciprocity Boundary-Element Method.

Linear stress analysis without body force can be easily solved by means of the boundary-element method. Some cases of linear stress analysis with body force can also be solved without a domain integral. However, domain integrals are generally necessary to solve the linear stress problem with complicated body force distribution. This paper shows that the three-dimensional linear stress problem with complicated body force distribution can be solved approximately without a domain integral by the triple-reciprocity boundary element method. In this method, the distribution of complicated body forces is interpolated by integral equation. A new computer program is developed and applied to some problems.

Stress analysis with arbitrary body force by triple-reciprocity BEM

Linear stress analysis without body force can be easily solved by means of the boundary element method. Some cases of linear stress analysis with body force can also be solved without a domain integral. However, domain integrals are generally necessary to solve the linear stress problem with arbitrary body forces. This paper shows that the linear stress problem with arbitrary body forces can be solved approximately without a domain integral by the triple-reciprocity boundary element method. In this method, the distribution of arbitrary body forces can be interpolated by the integral equation. A new computer program is developed and applied to several problems.

201 三重相反境界要素法を用いた任意の物体力を伴う三次元応力解析

201 三重相反境界要素法を用いた任意の物体力を伴う三次元応力解析

Stress Analysis with Arbitrary Body Force by Improved Multiple-Reciprocity Boundary-Element Method.

Linear stress analysis without body force can be easily solved by means of the boundary-element method. Some cases of linear stress analysis with body force can also be solved without a domain integral. However, domain integrals are generally necessary to solve the linear stress problem with complicated body forces. This paper shows that the linear stress problem with complicated body forces can be solved approximately without a domain integral by improved multiple-reciprocity boundary element method. In this method, the distribution of complicated body forces is interpolated by integral equation. A new computer program is develoed and applied to several problems.

Stress analysis with arbitrary body force by boundary element method

Linear stress analysis without body force can be easily carried out by means of the boundary element method. Some cases of linear stress analysis with body force can also be solved without the domain integral. However domain integrals are generally necessary to solve the linear stress problems with complicated body forces. This paper shows that the linear stress problems with complicated body forces can be solved approximately without the domain integral. In order to solve these problems, the domain is divided into small areas using contour lines of body force. In these areas, the distributions of body force are assumed approximately to satisfy the Laplace equation.