A parallel sparse algorithm targeting arterial fluid mechanics computations

Iterative solution of large sparse nonsymmetric linear equation systems is one of the numerical challenges in arterial fluid–structure interaction computations. This is because the fluid mechanics parts of the fluid + structure block of the equation system that needs to be solved at every nonlinear iteration of each time step corresponds to incompressible flow, the computational domains include slender parts, and accurate wall shear stress calculations require boundary layer mesh refinement near the arterial walls. We propose a hybrid parallel sparse algorithm, domain-decomposing parallel solver (DDPS), to address this challenge. As the test case, we use a fluid mechanics equation system generated by starting with an arterial shape and flow field coming from an FSI computation and performing two time steps of fluid mechanics computation with a prescribed arterial shape change, also coming from the FSI computation. We show how the DDPS algorithm performs in solving the equation system and demonstrate the scalability of the algorithm.     

Solution of elasticity problems by a self-adaptive mesh refinement technique for boundary element computation

A self-adaptive mesh refinement technique is developed for boundary element solutions of the two-dimensional Navier's equation. The method is based on error reduction and applied on the element and global level to estimate the residual error associated with each mesh. Results for several problems employing constant two-dimensional isotropic elastostatic boundary elements are presented.     

Anisotropic mesh refinement for problems with internal and boundary layers

The character of convection‐dominated, singularly perturbed boundary value problems requires their special numerical treatment in order to guarantee stability and resolve existing layers with acceptable accuracy. In addition to discretization methods particularly developed for this aim, recently more and more attention has been directed towards adapted triangulations of the computational domain. In this paper, an adaptive strategy based on an anisotropic refinement is developed for finite element methods. Starting from some a priori information about the location of layers, the so‐called hybrid meshes are constructed. By these meshes, the flexibility of unstructured meshes, good approximation properties in layers, and relatively simple rules for a posteriori anisotropic refinement are combined with each other. The efficiency of this procedure is demonstrated by selected numerical examples. Copyright © 1999 John Wiley & Sons, Ltd.     

A coupled BEM/scaled boundary FEM formulation for accurate computations in linear elastic fracture mechanics

Issues relating to the practical implementation of the coupled boundary element–scaled boundary finite element method are addressed in this paper. A detailed approach highlights fully the process of applying boundary conditions, including the treatment of examples in which the assumptions made in previous work are no longer valid. Verification of the method is undertaken by means of estimating stress intensity factors and comparing them against analytical solutions. The coupled algorithm shows good convergence properties. Issues relating to traction scaling, the use of discontinuous boundary elements, and the greater versatility of the coupled method over its constituent methods are highlighted.     

An HR-method of mesh refinement for boundary element method

This paper describes a mesh refinement technique for boundary element method in which the number of elements, the size of elements and the element end location are determined iteratively in order to obtain a user specified accuracy. The method uses L₁ norm as a measure of error in the density function and a grading function that ensures that error over each element is the same. The use of grading function along with L₁ norm makes the mesh refinement technique applicable to Direct and Indirect boundary element method formulation for a variety of boundary element method applications. Numerical problems in elastostatics, fracture mechanics, and bending of plate solved using Direct and Indirect method in which the density functions are approximated by Linear Lagrange, Quadratic Lagrange or Cubic Hermite polynomials validate the effectiveness of the proposed mesh refinement technique. © 1998 John Wiley & Sons, Ltd.     

Solution of the boundary layer equation for a power law fluid in magneto-hydrodynamics

Summary The problem of a non-Newtonian power law conducting fluid past a semi-finite plate, in the presence of an external electromagnetic field, is studied. The electric conductivity is taken as a function of the velocity. The method of expansion for a small parameter is used to obtain the solution and for a special form of the free stream velocity, the method of similarity solutions is used to obtain the exact solutions of the boundary layer equations associated with this problem in a closed form. It was found that the presence of the electromagnetic field reduces the velocity of the fluid.     

A refinement of the Karman-Pohlhausen integral method in boundary layer theory

A basis relation for boundary layer calculations has been obtained by double integration of the Prandtl equations. It is shown that this method leads to more accurate results than the Karman-Pohlhausen method.     

Error estimation and adaptive mesh refinement in boundary element method,an overview

Further to the previous review article (Adv Engng Software 19(1) (1994) 21–32), this paper reviews more recent studies on the same subject by citing more than one hundred papers.The adaptive mesh refinement process is composed of three processes; the error estimation, the adaptive tactics and the mesh refinement processes. Therefore, in this paper, the existing studies are classified and discussed according to the processes.The error estimation schemes are classified into the residual-type, the interpolation-type, the integral equation-type, the node sensitivity-type and the solution difference type. The mesh refinement schemes are classified into h-, p-, r-schemes and the others. The adaptive tactics are closely related to the mesh refinement schemes. Therefore, they are discussed individually.The discussion presented herein is an extension of the previous article and focuses our principal attention on the following points. Some interesting studies for the error estimation scheme are added; e.g. new schemes named as ‘nodal design sensitivity’, ‘hyper-singular residual type’ and ‘solution difference type’. Some studies for the adaptive tactics are added; e.g. the tactics based on the convergence property of the error, the extension of the extended error indicator to r- and hr-adaptive schemes and so on.     

Dynamic refinement and boundary contact forces in SPH with applications in fluid flow problems

This paper presents a general method for dynamic particle refinement in smoothed particle hydrodynamics (SPH). Candidate particles are split into several ‘daughter’ particles according to a given refinement pattern centred about the original particle. Through the solution of a non‐linear minimization problem the optimal mass distribution of the daughter particles is obtained so as to reduce the errors introduced to the underlying density field. This procedure necessarily conserves the mass of the system. Conservation of energy and momentum results are also discussed. Copyright © 2007 John Wiley & Sons, Ltd.     

Adaptive mesh refinement of the boundary element method for potential problems by using mesh sensitivities as error indicators

This paper presents a novel method for error estimation and h-version adaptive mesh refinement for potential problems which are solved by the boundary element method (BEM). Special sensitivities, denoted as mesh sensitivities, are used to evaluate a posteriori error indicators for each element, and a global error estimator. A mesh sensitivity is the sensitivity of a physical quantity at a boundary node with respect to perturbation of the mesh. The element error indicators for all the elements can be evaluated from these mesh sensitivities. Mesh refinement can then be performed by using these element error indicators as guides.The method presented here is suitable for both potential and elastostatics problems, and can be applied for adaptive mesh refinement with either linear or quadratic boundary elements. For potential problems, the physical quantities are potential and/or flux; for elastostatics problems, the physical quantities are tractions/displacements (or tangential derivatives of displacements). In this paper, the focus is on potential problems with linear elements, and the proposed method is validated with two illustrative examples. However, it is easy to extend these ideas to elastostatics problems and to quadratic elements.The computing for this research has been supported by the Cornell National Supercomputer Facility.     

Free convection boundary layer flow of a thermomicropolar fluid with stretch

The free convective boundary layer of a micropolar fluid with stretch has been studied numerically. The governing equations and boundary conditions are stated and solved in the case of a flow past a vertical flat plate. Steady state and transient developments of the flow field are given along with a detailed study of the effect of microstructure on heat transfer.     

A self-adaptive mesh refinement technique for boundary element solution of the Laplace equation.

A self-adaptive mesh refinement technique is developed for boundary element solutions of the two-dimensional Laplace equation. The method is based on error reduction and applied on the element and global level to estimate the error associated with each mesh. This adaptive technique is then utilized to analyze problems with and without singularities. Results employing constant two-dimensional boundary elements are presented.     

Separation of boundary layer in pulsating flows of incompressible fluid

Any equations obtained for determining the pulsation frequency at which the boundary layer separates have a function of the relative amplitude, the pipe diameter, and the average flow rate of the fluid.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 17, No. 2, pp. 332–336, August, 1969.     

On a linear steady-state system of equations in microcontinuum fluid mechanics

Galerkin representations and integral representations are obtained for the linearized system of coupled differential equations governing steady incompressible flow of a micropolar fluid. The special case of 2-dimensional Stokes flows is then examined and further representation formulae as well as asymptotic expressions, are generated for both the microrotation and velocity vectors. With the aid of these formulae, the Stokes Paradox for micropolar fluids is established.     

Solution of linear boundary inverse problems of heat conduction

New methods of solving linear one-dimensional boundary inverse problems of heat conduction are proposed; the methods are convenient for practical realization.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 34, No. 3, pp. 529–535, March, 1978.     

Effects of fluid layer at micropolar orthotropic boundary surface

Effects of a fluid layer at a micropolar orthotropic elastic solid interface to a moving point load have been studied. After using the Fourier transform an eigen value approach has been employed to solve the problem. The displacement, microrotation and stress components for a micropolar orthotropic elastic solid so obtained in the physical domain are computed numerically by applying the numerical inversion technique. Micropolarity and anisotropy effects along with that of the depth of the fluid layer on various expressions have been depicted graphically for a particular model. Some special cases of interest have been presented     

Comments on the solutions of boundary value problems in non-Newtonian fluid mechanics

Summary The problem of the availability of boundary conditions in non-Newtonian fluid mechanics is briefly reviewed. The flow with suction over a flat plate of differential, integral, and hierarchy types of fluids is studied. It is concluded that while differential and integral models, with physically motivated conditions, give unique solutions, the hierarchy model gives rise to non-uniqueness. An alternate method which provides a better approximate solution for a hierarchy model is suggested.     

On fluid dynamic drag reduction in some boundary layer flows

Summary The problem of boundary layer flow on a flat plate with injection and a constant velocity opposite in direction to that of the uniform mainstream is analyzed. It is shown that the solution of this boundary layer problem not only depends on the ratio of the velocity of the plate to the velocity of the free stream (), but also on the injection velocity parameter (C). It is also shown that there exists a range of values of andC for which the differential equations associated with the boundary layer problems admit analytic solutions. The critical values of andC are obtained numerically and their significance in drag reduction is discussed.     

Onset of thermal instability in a horizontal layer of fluid with modulated boundary temperatures

Thermal instability in a horizontal layer of fluid, with the boundary temperatures modulated sinusoidally in time, is studied. The amplitude of modulation is assumed small and is used as an expansion parameter. It is shown that an exact solution can be obtained, even when the boundaries are considered to be rigid. When only the lower boundary temperature is modulated, for small values of the Prandtl number modulation is always stabilizing, while for large values it can be stabilizing or destabilizing depending on the modulation frequency. When both boundary temperatures are modulated in phase, modulation is destabilizing for low modulation frequency, but for higher modulation frequency stabilization occurs for low values of the Prandtl number. When the two boundary temperatures are modulated out of phase the modulation always has a stabilizing effect.