Fictitious Body Forces Research Articles

The localized method of fundamental solutions for 2D and 3D second-order nonlinear boundary value problems

In this paper, a new framework for the numerical solutions of general nonlinear problems is presented. By employing the analog equation method, the actual problem governed by a nonlinear differential operator is converted into an equivalent problem described by a simple linear equation with unknown fictitious body forces. The solution of the substitute problem is then obtained by using the localized method of fundamental solutions, where the fictitious nonhomogeneous term is approximated using the dual reciprocity method using the radial basis functions. The main difference between the classical and the present localized method of fundamental solutions is that the latter produces sparse and banded stiffness matrix which makes the method very suitable for large-scale nonlinear simulations, since sparse matrices are much cheaper to inverse at each iterative step of the Newton's method. The present method is simple in derivation, efficient in calculation, and may be viewed as a completive alternative for nonlinear analysis, especially for large-scale problems with complex-shape geometries. Preliminary numerical experiments involving second-order nonlinear boundary value problems in both two- and three-dimensions are presented to demonstrate the accuracy and efficiency of the present method.

A reduced unified continuum formulation for vascular fluid–structure interaction

We recently derived the unified continuum and variational multiscale formulation for fluid–structure interaction (FSI) using the Gibbs free energy as the thermodynamic potential. Restricting our attention to vascular FSI, we now reduce this formulation in arbitrary Lagrangian–Eulerian (ALE) coordinates by adopting three common modeling assumptions for the vascular wall. The resulting semi-discrete formulation, referred to as the reduced unified continuum formulation, achieves monolithic coupling of the FSI system in the Eulerian frame through a simple modification of the fluid boundary integral. While ostensibly similar to the semi-discrete formulation of the coupled momentum method introduced by Figueroa et al., its underlying derivation does not rely on an assumption of a fictitious body force in the elastodynamics sub-problem and therefore represents a direct simplification of the ALE method. Furthermore, uniform temporal discretization of the entire FSI system is performed via the generalized-α scheme. In contrast to the predominant approach yielding only first-order accuracy for pressure, we collocate both pressure and velocity at the intermediate time step to achieve uniform second-order temporal accuracy. In conjunction with quadratic tetrahedral elements, our methodology offers higher-order temporal and spatial accuracy for quantities of clinical interest, including pressure and wall shear stress. Furthermore, without loss of consistency, a segregated predictor multi-corrector algorithm is developed to preserve the same block structure as for the incompressible Navier–Stokes equations in the implicit solver’s associated linear system. Block preconditioning of a monolithically coupled FSI system is therefore made possible for the first time. Compared to alternative preconditioners, our three-level nested block preconditioner, which achieves improved representation of the Schur complement, demonstrates robust performance over a wide range of physical parameters. We present verification of our methodology against Womersley’s deformable wall theory and additionally develop practical modeling techniques for clinical applications, including tissue prestressing. We conclude with an assessment of our combined FSI technology in two patient-specific cases.

Simulating disturbance frequency effect on heat transfer enhancement in wavy condensate films

It is of great significance to investigate microscopic mechanisms of heat transfer enhancement in wavy condensate films which are frequently encountered in nuclear power plants. A numerical model, in which the wavy interface is tracked by a non-orthogonal coordinate transformation, is developed to study the disturbance frequency effect on heat transfer enhancement along a vertical wall. A sinusoidal disturbance is introduced by a fictitious body force to sustain waves. A disturbance frequency of 16 Hz associated with the critical film Reynolds number is numerically determined by analyzing the neutral stability curve. Compared with other disturbance frequencies, the wave effect on local heat transfer coefficients is the most significant at 16 Hz. The minimum time-averaged thickness and the maximum proportion of substrate film length in the wave length at 16 Hz proves that the film thinning effect is determined by the length of the film substrate rather than the substrate film thickness. The more notable variation of the normal velocity close to the wall at the wave front at 16 Hz accounts for the convection effect of heat transfer enhancement.

Microscopic mechanisms of wave effect on heat transfer enhancement in condensate films

In order to deeply understand the inherent links between hydrodynamic behaviors and heat transfer enhancement in wavy condensate films, a numerical model using a non-orthogonal coordinate transformation was developed to simulate the evolution of wavy films. A sinusoidal disturbance was introduced by a fictitious body force to sustain waves. Two major pressure gradients are established at the wave front. One parallel to the stream-wise direction accelerates fluids and the other normal to the wall increases the normal velocity. The temperature profiles prove that both convection and conduction contribute microscopic mechanisms to heat transfer enhancement in the condensate films. The normal velocity close to the wall significantly influences the temperature gradient at the wall. The predicted heat transfer coefficient for the wavy film was found to be 30% greater than the Nusselt solution for the given parameters. Under the given conditions, the effects of film thinning and convection account for 6% and 24% of the heat transfer enhancement, respectively.

The analog equation integral formulation for plane piezoelectric media

In this paper, the two-dimensional piezoelectricity is modelled using a boundary integral formulation based on its corresponding Analog Equation. The problem is transformed into three uncoupled Poisson׳s equations with unknown fictitious body forces terms. The multiquadric radial basis function is used to approximate the fictitious body forces in the particular solutions. The problem is solved by satisfying the governing differential operator and the boundary conditions. A scaling process is used to enhance the numerical behaviour of the obtained system of equations. The formulation is mathematically simpler than formerly proposed BEM formulations and its validity, applicability and accuracy are demonstrated through various numerical examples.

三重相反境界要素法による不均質材のメッシュレス応力解析

In general, internal cells are required to solve nonhomogeneous elastic problems using a conventional boundary element method (BEM). However, in this case, the merit of the BEM, which is an ease of data preparation, is lost. In this study, it is shown that two-dimensional nonhomogeneous elastic problems of functionally gradient materials can be solved without the use of internal cells, using the triple-reciprocity BEM. The triple-reciprocity BEM can be applied to thermo-elastoplastic problems with arbitrary heat generation and to three-dimensional elastoplastic problems without internal cells. In this paper, Young's modulus and Poisson's ratio are variable in nonhomogeneous elastic materials. In this method, boundary elements and arbitrary internal points are used. The distribution of fictitious body force generated by a nonhomogeneous material is interpolated using boundary integral equations. This interpolation corresponds to a thin plate spline. In this paper, elastic analysis is carried out for laminated materials, porous materials and a particle-dispersed composite as special cases of functionally gradient materials. In the case of composite materials or a layer structure, the distribution of Lame's constant is not continuous. However, the same interpolation method can be used for special case of nonhomogeneous material. A new computer program is developed and applied to several nonhomogeneous elastic problems to clearly demonstrate the theory.

Recovery of equilibrium on star patches from conforming finite elements with a linear basis

SUMMARY A previous technique for recovering equilibrated stresses from compatible finite element models of structural mechanics problems is extended to cover those cases where the partitioned loads applied to star patches are not initially balanced, regarding rotational equilibrium. The residual moments are removed by adding a suitable corrective stress field to the compatible one before deriving the fictitious body forces. Corrective stress fields are determined by solving another set of local problems based on subdomains that each contain elements forming a neighbourhood of a loaded kernel element. The conditions for the existence of a solution of these problems are studied for simplicial elements. The parameters that control the extended technique are assessed from numerical tests on a variety of two-dimensional linear elastic problems based on constant strain elements for the compatible solutions. These tests are presented in the context of computing bounds on quantities of interest such as total strain energy and local reactions and displacements. Copyright © 2011 John Wiley & Sons, Ltd.

Critical review of conservation equations for two-phase flow in the U.S. NRC TRACE code

The field equations for two-phase flow in the computer code TRAC/ RELAP Advanced Computational Engine or TRACE are examined to determine their validity, their capabilities and limitations in resolving nuclear reactor safety issues. TRACE was developed for the NRC to predict thermohydraulic phenomena in nuclear power plants during operational transients and postulated accidents. TRACE is based on the rigorously derived and well-established two-fluid field equations for 1-D and 3-D two-phase flow. It is shown that: (1) The two-fluid field equations for mass conservation as implemented in TRACE are wrong because local mass balances in TRACE are in conflict with mass conservation for the whole reactor system, as shown in Section 3.1. (2) Wrong equations of motion are used in TRACE in place of momentum balances, compromising at branch points the prediction of momentum transfer between, and the coupling of, loops in hydraulic networks by impedance (form loss and wall shear) and by inertia and thereby the simulation of reactor component interactions. (3) Most seriously, TRACE calculation of heat transfer from fuel elements is incorrect for single and two-phase flows, because Eq. (3-4) of the TRACE Manual is wrong (see Section 5.2). (4) Boundary conditions for momentum and energy balances in TRACE are restricted to flow regimes 1 1 For classification of two-phase flow regimes see Delhaye et al. (1981, p. 47). with single-phase wall contact because TRACE lacks constitutive relations for solid–fluid exchange of momentum and heat in prevailing flow regimes. Without a quantified assessment of consequences from (3) to (4), predictions of phasic fluid velocities, fuel temperatures and important safety parameters, e.g., peak clad temperature, are questionable. Moreover, TRACE cannot predict 3-D single- or two-phase flows because: (5) incorrectly averaged equations are used for 3-D predictions, (6) fluid shear is ignored but needed to predict counter-current flows with the two-fluid model, and (7) fictitious body forces and fictitious distributed mass and heat sources are used to replace contact forces at the wall, mass injection and wall heat fluxes in 3-D mass, momentum and energy equations for both phases. No modeling error estimates are given in the TRACE Manual. (8) Imposed perfect mixing in every control volume causes artificial damping and disables TRACE to track reliably propagation of thermal and kinematic disturbances, thereby adversely affecting the prediction of nuclear fuel temperature. (9) According to its manual, TRACE relies on numerical diffusion far greater than physical dissipation by turbulence, to solve TRACE's ill-posed field equations and to compensate for missing fluid shear. TRACE with scale-sensitive numerical diffusion is tuned to match data from small-scale experiments. TRACE is therefore not reliable for analyzing full-size power plants. Numerical diffusion could also reduce the ability of TRACE to predict the onset of instability for reactors or test facilities entering large power and flow oscillations. Agreements of TRACE code results with experimental data from test facilities with isolated full-size components or with data from reduced-size integral-effect tests may be achieved by adjusting numerous coefficients, some of them scale-dependent (i.e., by tuning with documented non-physical coefficients in momentum balances, by spurious time-averaging (through time step adjustment) of algebraic boundary conditions, by built-in, non-physical space grid tuning in momentum balances and in mixture level predictions). Such results may not be applied reliably to full-size systems with realistic interactions between reactor components. TRACE may not be applicable to predict the outcome of postulated accidents for which validations are not possible (e.g., Anticipated Transients without Scram). New experimental techniques are needed first for closure of the two-fluid model. A new approach to TRACE modeling is needed. Recommendations are given for improved code documentation.

Recovery of equilibrium on star patches using a partition of unity technique

AbstractA technique for recovering equilibrated element stresses is developed for finite element models of structural mechanics problems. The data for the method consist of the prescribed loading and the stress and displacement fields resulting from a conventional compatible finite element model. Local problems are defined for each star of elements, via the introduction of fictitious body forces and strains. These problems can be solved independently for equilibrium in a process that can be easily parallelized. The quality of the solutions is assessed, for two‐dimensional linear elastic problems, by using them to compute bounds of the error of the finite element solutions, in terms of both global and local quantities of interest. Copyright © 2009 John Wiley & Sons, Ltd.

Building resolving large-eddy simulations and comparison with wind tunnel experiments

We perform large-eddy simulations (LES) of the flow past a scale model of a complex building. Calculations are accomplished using two different methods to represent the edifice. The first method employs the standard Gal-Chen and Somerville terrain-following coordinate transformation, common in mesoscale atmospheric simulations. The second method uses an immersed boundary approach, in which fictitious body forces in the equations of motion are used to represent the building by attenuating the flow to stagnation within a time comparable to the time step of the model. Both methods are implemented in the same hydrodynamical code (EULAG) using the same nonoscillatory forward-in-time (NFT) incompressible flow solver based on the multidimensional positive definite advection transport algorithms (MPDATA). The two solution methods are compared to wind tunnel data collected for neutral stratification. Profiles of the first- and second-order moments at various locations around the model building show good agreement with the wind tunnel data. Although both methods appear to be viable tools for LES of urban flows, the immersed boundary approach is computationally more efficient. The results of these simulations demonstrate that, contrary to popular opinion, continuous mappings such as the Gal-Chen and Somerville transformation are not inherently limited to gentle slopes. Calculations for a strongly stratified case are also presented to point out the substantial differences from the neutral boundary layer flows.

A deterministic model for the sublayer streaks in turbulent boundary layers for application to flow control

We present a relatively simple, deterministic, theoretical model for the sublayer streaks in a turbulent boundary layer based on an analogy with Klebanoff modes. Our approach is to generate the streamwise vortices found in the buffer layer by means of a vorticity source in the form of a fictitious body force. It is found that the strongest streaks correspond to a spanwise wavelength that lies within the range of the experimentally observed values for the statistical mean streak spacing. We also present results showing the effect of streamwise pressure gradient, Reynolds number and wall compliance on the sublayer streaks. The theoretical predictions for the effects of wall compliance on the streak characteristics agree well with experimental data. Our proposed theoretical model for the quasi-periodic bursting cycle is also described, which places the streak modelling in context. The proposed bursting process is as follows: (i) streamwise vortices generate sublayer streaks and other vortical elements generate propagating plane waves, (ii) when the streaks reach a sufficient amplitude, they interact nonlinearly with the plane waves to produce oblique waves that exhibit transient growth, and (iii) the oblique waves interact nonlinearly with the plane wave to generate streamwise vortices; these in turn generate the sublayer streaks and so the cycle is renewed.

Analysis of thermal-elastic stress of wheel-rail in rolling-sliding contact

A coupling thermo-mechanical model of wheel/rail in rolling-sliding contact is put forward using finite element method. The normal contact pressure is idealized as the Hertzian distribution, and the tangential force presented by Carter is used. In order to obtain thermal-elastic stress, the ther-mal-elastic plane stress problem is transformed to an elastic plane stress problem with equivalent fictitious thermal body force and fictitious boundary distributed force. The temperature rise and ther-mal-elastic stress of wheel and rail in rolling-sliding are analyzed. The non-steady state heat transfer between the contact surfaces of wheel and rail, heat-convection and radiation between the wheel/rail and the ambient are taken into consideration. The influences of the wheel rolling speed and wear rate on friction temperature and thermal-elastic stress are investigated. The results show the following: ① For rolling-sliding case, the thermal stress in the thin layer near the contact patch due to the friction temperature rise is severe. The higher rolling speed leads to the lower friction temperature rise and thermal stress in the wheel; ② For sliding case, the friction temperature and thermal stress of the wheel rise quickly in the initial sliding stage, and then get into a steady state gradually. The expansion of the contact patch, due to material wear, can affect the friction temperature rise and the thermal stress during wear process. The higher wear rate generates lower stress. The results can help under-stand the influence of friction temperature and thermal-elastic stress on wheel and rail damage.

A meshless method for generalized linear or nonlinear Poisson-type problems

This paper presents a meshless method, based on coupling virtual boundary collocation method (VBCM) with the radial basis functions (RBF) and the analog equation method (AEM), to analyze generalized linear or nonlinear Poisson-type problems. In this method, the AEM is used to construct equivalent equations to the original differential equation so that a simpler fundamental solution of the Laplacian operator, instead of other complicated ones which are needed in conventional BEM, can be employed. While global RBF is used to approximate fictitious body force which appears when the analog equation method is introduced, and VBCM are utilized to solve homogeneous solution based on the superposition principle. As a result, a new meshless method is developed for solving nonlinear Poisson-type problems. Finally, some numerical experiments are implemented to verify the efficiency of the proposed method and numerical results are in good agreement with the analytical ones. It appears that the proposed meshless method is very effective for nonlinear Poisson-type problems.

AN EFFICIENT PSEUDO THREE-DIMENSIONAL FINITE ELEMENT MODEL: PRESENTATION AND ANALYSIS OF TWO SOIL/FOUNDATION INTERACTION PROBLEMS

A pseudo-three-dimensional numerical model has been developed for the analysis of full 3D soil problems under combined loading. The procedure called Vertical Slices Model takes advantage of finite element (FE) 2D numerical solutions in plane stress for building approximate 3D solutions by replacing the inter-slice interactions by fictitious body forces. Continuum slices are successively analyzed by the combination of the explicit 2D finite element (FE) method and finite difference (FD) method in iterative process. The three-dimensional aspect of the considered problem is preserved by satisfying the continuity of shear stresses developed at the inter-slices. The theory of the vertical slices model is developed first, and then encoded in a Fortran computer program. Next, the prediction capabilities of this program are illustrated with two classical geotechnical applications, namely; the laterally and the axially loaded single piles embedded in homogeneous and non-homogeneous elastic soils. Although approximate, the model proved its ability to capture the behavior of the two boundary value problems. Then, in terms of stiffness factors the approach is used to predict the behavior of an embedded rigid square footing and a pile under combined loading in a half-space where the stiffness shows a power law variation with depth.

Finite element vertical slices model: Validation and application to an embedded square footing under combined loading

This paper describes an efficient formulation for the analysis of three-dimensional (3-D) soil/structure interaction problems. The technique which is called vertical slices model (VSM) subdivides the structure and the surrounding soil into vertical slices. It takes profit of finite element 2-D numerical solutions in plane stress for building approximate 3-D solutions by replacing the inter-slice interactions by fictitious body forces. The 2-D finite element and finite difference methods are combined in a relatively simple and compact procedure, and applied iteratively to each vertical slice. The 3-D aspect of the considered problem is kept by satisfying the continuity of shear stresses developed at the inter-slices. First, the theory of the vertical slices model is presented and then a Fortran computer program is elaborated. Next, the capabilities of this program are illustrated by applying the approach to the problems of both laterally and axially loaded piles founded in both homogeneous and non-homogeneous soils. A third problem of a rigid rectangular foundation resting on top of a homogeneous elastic stratum has also been investigated. Comparisons with other numerical solutions show good agreement and consequently they prove that the model is successfully able to capture the behavior of the three boundary value problems. Finally, in terms of stiffness factors the method is used to predict the behavior of an embedded square footing under combined loading, in a half-space where the stiffness exhibits a power-law variation with depth.

Seismic Waves in a Laterally Inhomogeneous Layered Medium, Part I: Theory

Seismic waves in a layered half-space with lateral inhomogeneities, generated by a buried seismic dislocation source, are investigated in these two consecutive papers. In the first paper, the problem is formulated and a corresponding approach to solve the problem is provided. Specifically, the elastic parameters in the laterally inhomogeneous layer, such as P and S wave speeds and density, are separated by the mean and the deviation parts. The mean part is constant while the deviation part, which is much smaller compared to the mean part, is a function of lateral coordinates. Using the first-order perturbation approach, it is shown that the total wave field may be obtained as a superposition of the mean wave field and the scattered wave field. The mean wave field is obtainable as a response solution for a perfectly layered half-space (without lateral inhomogeneities) subjected to a buried seismic dislocation source. The scattered wave field is obtained as a response solution for the same layered half-space as used in the mean wave field, but is subjected to the equivalent fictitious distributed body forces that mathematically replace the lateral inhomogeneities. These fictitious body forces have the same effects as the existence of lateral inhomogeneities and can be evaluated as a function of the inhomogeneity parameters and the mean wave fleld. The explicit expressions for the responses in both the mean and the scattered wave fields are derived with the aid of the integral transform approach and wave propagation analysis.

Seismic Waves in a Laterally Inhomogeneous Layered Medium, Part II: Analysis

Seismic wave scattering representation for the layered half-space with lateral inhomogeneities subjected to a seismic dislocation source has been formulated in the companion paper with the use of first-order perturbation (Born-type approximation) technique. The total wave field is obtained as a superposition of the mean and the scattered wave fields, which are generated, respectively, by a series of double couples of body forces equivalent to the seismic dislocation source and by fictitious body forces equivalent to the existence of the lateral inhomogeneities in the layered half-space. The responses in both the mean and the scattered wave fields are found with the aid of an integral transform technique and wave propagation analysis. The characteristics of the scattered waves and their effects on the mean waves or corresponding induced ground and/or underground mean responses are investigated in this paper. In particular, coupling phenomena between P-SV and SH waves and wave number shifting effects between the mean and the scattered wave responses are presented in detail. With the lateral inhomogeneities being assumed as a homogeneous random field, a qualitative analysis is provided for estimating the effects of the lateral inhomogeneities on the ground motion, which is related to a fundamental issue: whether a real earth medium can or cannot be approximately considered as a laterally homogeneous layer. The effects of the lateral inhomogeneities on the ground motion time history are also presented as a quantitative analysis. Finally, a numerical example is carried out for illustration purposes.

A method for thermo-mechanical analysis of steady state dynamic crack growth

Cracks propagating at high speeds in metals generate heat at their tips which cannot be conducted away in the very short time span of crack growth. The effect of heating on the crack tip stress fields and on dynamic fracture toughness is analyzed using a method based on the theory of generalized analytic functions. In this method all deviations from a linear elastic, homogeneous strain field are represented as fictitious body forces, allowing one to model inelastic and thermal effects. Numerical computations of the crack tip stress and strain fields are performed assuming steady-state, plane stress, mode-I dynamic crack growth with a fixed crack tip temperature field, but with temperature and strain rate sensitive mechanical properties. The relationship between dynamic stress intensity factor and crack velocity has been determined using competing ductile and brittle fracture criteria. A careful analysis of the trailing wake has been performed to study its effect on the crack tip stress field.

Thermoviscoplastic analysis of fibrous periodic composites by the use of triangular subvolumes

The non-linear viscoplastic behavior of fibrous periodic composites is analyzed by discretizing the unit cell into triangular subvolumes. A set of these subvolumes can be configured by the analyst to construct a representation for the unit cell of a periodic composite. In each step of the loading history the total strain increment at any point is governed by an integral equation which applies to the entire composite. A Fourier series approximation allows the incremental stresses and strains to be determined within a unit cell of the periodic lattice. The non-linearity arising from the viscoplastic behavior of the constituent materials comprising the composite is treated as a fictitious body force in the governing integral equation. Specific numerical examples showing the stress distributions in the unit cell of a fibrous tungsten/copper metal-matrix composite under viscoplastic loading conditions are given. The stress distribution resulting in the unit cell when the composite material is subjected to an overall transverse stress loading history perpendicular to the fibers is found to be highly heterogeneous, and typical homogenization techniques based on treating the stress and strain distributions within the constituent phases as homogeneous result in large errors under inelastic loading conditions.

Accuracy of the generalized self-consistent method in modelling the elastic behaviour of periodic composites

Local stress and strain fields in the unit cell of an infinite, two-dimensional, periodic fibrous lattice have been determined by an integral equation approach. The effect of the fibres is assimilated to an infinite two-dimensional array of fictitious body forces in the matrix constituent phase of the unit cell. By subtracting a volume averaged strain polarization term from the integral equation we effectively embed a finite number of unit cells in a homogenized medium in which the overall stress and strain correspond to the volume averaged stress and strain of the constrained unit cell. This paper demonstrates that the zeroth term in the governing integral equation expansion, which embeds one unit cell in the homogenized medium, corresponds to the generalized self-consistent approximation. By comparing the zeroth term approximation with higher order approximations to the integral equation summation, both the accuracy of the generalized self-consistent composite model and the rate of convergence of the integral summation can be assessed. Two example composites are studied. For a tungsten/copper elastic fibrous composite the generalized self-consistent model is shown to provide accurate, effective, elastic moduli and local field representations. The local elastic transverse stress field within the representative volume element of the generalized self-consistent method is shown to be in error by much larger amounts for a composite with periodically distributed voids, but homogenization leads to a cancelling of errors, and the effective transverse Young’s modulus of the voided composite is shown to be in error by only 23% at a void volume fraction of 75%.