Spline-based meshfree method with extended basis

In this work, an extension has been performed on the analysis basis of spline-based meshfree method (SBMFM) to stabilize its solution. The potential weakness of the SBMFM is its numerical instability from using regular grid background mesh. That is, if an extremely small trimmed element is produced by the trimming curves that represent boundaries of the analysis domain, it can induce an excessively large condition number in global system matrix. To resolve the instability problem, the extension technique of the weighted extended B-spline (WEB-spline) is implemented in the SBMFM. The basis functions with very small trimmed supports are extrapolated by neighboring basis functions with some special scheme so that those basis functions can be condensed in the solution process. In order to impose essential boundary conditions in the SBMFM with extended basis, Nitsche's method is implemented. Using numerical examples, the presented SBMFM with extended basis is shown to be valid and effective. Moreover, the condition number of the system is well-managed guaranteeing the stability of the numerical analysis.     

A least-squares/penalty method for distributed optimal control problems for Stokes equations

The purpose of this paper is to construct an unconstrained optimal control problem by using a least-squares approach for the constrained distributed optimal control problem associated with incompressible Stokes equations. The constrained equations are reformulated to the equivalent first-order system by introducing vorticity, and then the least-squares functional corresponding to the system is enforced via a penalty term to the objective functional. The existence of a solution of the unconstrained optimal control problem is proved, and the convergence of this solution to that of unpenalized one is demonstrated as the penalty parameter tends to zero. Finite element approximations with error estimates are studied, and the relevant computational experiments are presented.     

Algorithm for analysing optical flow based on the least-squares method

An algorithm for analysing the optical flow field due to a camera moving through a rigid environment is presented. It relies on a least-squares estimate of the errors associated with possible values for the translational velocity. If the surfaces in view are irregular then some of the parameters of the motion can be obtained more accurately than the others in the presence of noise.     

Apparent slip arising from Stokes shear flow over a bidimensional patterned surface

A mathematical model is presented for the problem of apparent slip arising from Stokes shear flow over a composite surface featuring mixed boundary conditions on the microscale. The surface can be composed of a bidimensional array of solid areas placed on an otherwise no-shear surface corresponding to an envelope over the tops of posts, or no-shear areas placed on an otherwise solid surface corresponding to an envelope over the tops of holes. Posts and holes of circular or square cross section, and solid areas of no-slip or partial-slip types are studied. Following some previously proposed scaling laws, the effective slip length is expressed as a certain function of the solid fraction for some specific cases. More refined equations based on linear regression of the computed results are obtained for these cases. Amounts of slippage arising from these bidimensional patterns are compared with those from the one-dimensional patterns of grooves/grates. It is also shown that a larger slip length can result from an arrangement where the pitch is larger in the spanwise direction than in the streamwise direction.     

A lattice Boltzmann method for immiscible two-phase Stokes flow with a local collision operator

We propose a lattice Boltzmann model for immiscible two-phase Stokes flow with a local collision operator. The model is based on two different lattice Boltzmann automata, one for the flow field and one for an indicator function for the two different phases. The model is described in detail and verified by the following test-cases: a static bubble for the surface tension, a closed capillary tube for the contact angle and two phase flow in a concentric annulus for the viscosity ratio. In the appendix an asymptotic analysis for the derivation of the two-phase Stokes equation is given.     

Enriched meshfree collocation method with diffuse derivatives for elastic fracture

A meshfree collocation method with intrinsic enrichment for solving elastic crack problems is presented. A diffuse derivative approximation is applied in conjunction with an intrinsic enrichment of the near-tip asymptotic fields and a polynomial basis. These consistent diffuse derivatives of the approximation do not require computing the derivative of the weight function or that of the moment matrix. The local behavior of the near-tip stresses is successfully captured so that the stress intensity factors can be accurately computed. Numerical experiments demonstrate the accuracy and robustness of the method for solving elastic fracture problems.     

A fully Lagrangian mixed discrete least squares meshfree method for simulating the free surface flow problems

This paper presents a fully Lagrangian mixed discrete least squares meshfree (MDLSM) method for simulating the free surface problems. In the proposed method, the mass and momentum conservation equations are first discretized in time using the projection method. The resulting pressure Poisson equation is then re-written in the form of three first-order equations in terms of the pressure field and its first-order derivatives. The mixed discrete least squares meshless method is then used to solve this system of equations and simultaneously calculate the pressure field and its gradients. The advantage of the proposed Lagrangian MDLSM is twofold. First, the pressure gradients are directly computed and, therefore, they enjoy higher accuracy than those calculated in the conventional DLSM via a post-processing method. The more accurate pressure gradient will in turn lead to more accurate velocity field when used in the momentum equations. Second, the proposed Lagrangian MDLSM method can be more efficient than the corresponding original Lagrangian DLSM method, for the specific number of nodes, since the costly calculation of the shape function second derivatives required for solving the pressure Poison equation are avoided in each time step of the simulation. Several free surface problems are solved and the results are presented and compared to those of DLSM method. The results indicate the superior efficiency and accuracy of the proposed Lagrangian MDLSM method compared to those of the existing Lagrangian DLSM method in the literature.

     

Gridless spectral algorithm for Stokes flow with moving boundaries

A gridless, spectrally-accurate algorithm for the Stokes flow with moving boundaries is presented. The algorithm uses fixed computational domain with boundaries of the flow domain moving inside the computational domain. The spatial discretization is based on the Fourier expansions in the streamwise direction and the Chebyshev expansions in the transverse direction. Temporal discretization uses one- and two-steps implicit formulations. The boundary conditions on the moving boundaries are imposed using the immersed boundary conditions concept. Numerical tests confirm the spectral accuracy in space and theoretically-predicted accuracy in time. Different variants of the solution procedure are presented and their relative advantages are discussed.     

Space-time least-squares finite element method for convection-reaction system with transformed variables

We present a method to solve a convection-reaction system based on a least-squares finite element method (LSFEM). For steady-state computations, issues related to recirculation flow are stated and demonstrated with a simple example. The method can compute concentration profiles in open flow even when the generation term is small. This is the case for estimating hemolysis in blood. Time-dependent flows are computed with the space-time LSFEM discretization. We observe that the computed hemoglobin concentration can become negative in certain regions of the flow; it is a physically unacceptable result. To prevent this, we propose a quadratic transformation of variables. The transformed governing equation can be solved in a straightforward way by LSFEM with no sign of unphysical behavior. The effect of localized high shear on blood damage is shown in a circular Couette-flow-with-blade configuration, and a physiological condition is tested in an arterial graft flow.     

基于最小二乘的正交距离拟合方法研究

对点云进行拟合以获得信息模型是许多工程应用领域的一个核心问题,其目的是根据设计模型和规范的要求,高精度地对三维部件、工件进行检查、检核。在最小二乘原理的基础上,阐述了一种新的拟合方法--几何正交距离拟合,运用这种方法的算法处理标准几何体测量数据,并与一般代数拟合方法处理结果进行比较,分析了同一几何体运用不同方法对拟合结果的影响。得出在测量点误差较大的情况下和进行曲线曲面拟合时,几何正交距离拟合方法优于一般代数拟合方法。     

Fuzzy bounded least-squares method for the identification of linearsystems

This paper presents the fuzzy bounded least-squares method which uses both linguistic information and numerical data to identify linear systems. This method introduces a new type of fuzzy system, i.e., a fuzzy interval system. The steps in the method are as follows: 1) to utilize all the available linguistic information to obtain a fuzzy interval system and then to use the fuzzy interval system to give the admissible model set (i.e., the set of all models which are acceptable and reasonable from the point of view of linguistic information); 2) to find a model in the admissible model set which best fits the available numerical data. It is shown that such a model can be obtained by a quadratic programming approach. By comparing this method with the least-squares method, it is proved that the model obtained by this method fits a real system better than the model obtained by the least-squares method. In addition, this method also checks the adequacy of linear models for modeling a given system during the identification process and can help one to decide whether it is necessary to use nonlinear models     

Floating grid characteristics method for unsteady flow over a mobile bed

In this paper, we present a new application of the method of characteristics (MOC) for the 1D simulation of unsteady flow over a mobile bed. The method is based on the floating grid-of-characteristics approach, which here is amended to take account of the more complex structure of the field of characteristics inherent in the morphodynamics (compared to those for hydro- or gas dynamics). Due to the relative simplicity of the algorithm the technique is particularly well suited to waves moving into a downstream reach that is initially dry. As with all MOC schemes the approach is also attractive because it reveals the underlying (morpho)dynamics as the solution progresses. The mathematical model considered here is based on the depth averaged non-linear shallow water equations directly coupled to a bed update (Exner) equation. Closure is obtained using a simple sediment flux relation. The technique described allows the study of fully coupled sediment transport problems and the associated hydro- and morphodynamic evolution. Numerical results are presented for the evolution of a sand dune in a tidal flow as well as the wet-dry dam-break problem on both a fixed and mobile bed. Results compare extremely well with the analytical or quasi-analytical solutions for the problems presented.     

Application of the parallel BDDC preconditioner to the Stokes flow

A parallel implementation of the Balancing Domain Decomposition by Constraints (BDDC) method is described. It is based on formulation of BDDC with global matrices without explicit coarse problem. The implementation is based on the MUMPS parallel solver for computing the approximate inverse used for preconditioning. It is successfully applied to several problems of Stokes flow discretized by Taylor–Hood finite elements and BDDC is shown to be a promising method also for this class of problems.     

Incompressible flow computations over moving boundary using a novel upwind method

In this paper a novel method for simulating unsteady incompressible viscous flow over a moving boundary is described. The numerical model is based on a 2D Navier–Stokes incompressible flow in artificial compressibility formulation with Arbitrary Lagrangian Eulerian approach for moving grid and dual time stepping approach for time accurate discretization. A higher order unstructured finite volume scheme, based on a Harten Lax and van Leer with Contact (HLLC) type Riemann solver for convective fluxes, developed for steady incompressible flow in artificial compressibility formulation by Mandal and Iyer (AIAA paper 2009-3541), is extended to solve unsteady flows over moving boundary. Viscous fluxes are discretized in a central differencing manner based on Coirier’s diamond path. An algorithm based on interpolation with radial basis functions is used for grid movements. The present numerical scheme is validated for an unsteady channel flow with a moving indentation. The present numerical results are found to agree well with experimental results reported in literature.