Piecewise Continuous Representation Research Articles

A mixed MLS and Hermite-type MLS method for buckling and postbuckling analysis of thin plates

In the present paper, a new procedure, that combines the Asymptotic Numerical Method (ANM) with the Hermite-Type Moving Least Squares (HMLS) meshless method, is proposed to study the bending, buckling and postbuckling behaviors of Kirchhoff–Love thin plate model. The MLS shape functions are used to approximate the in-plane displacements, while Hermite-type MLS shape functions are used to approximate the transversal displacement and its derivatives. To overcome the difficulty in the imposition of the essential boundary conditions, we use the Lagrange multiplier method. The ANM is adopted in order to derive the results from the global nonlinear system of equations obtained from the principle of the stationarity condition of the total potential energy. This algorithm is based on the coupling of the Taylor series expansion and a continuation process. This approach allows the construction of the solution curve step by step thanks to a piecewise continuous representation. The continuation technique is performed by using the arc-length method. The effectiveness of the proposed coupling is brought out through illustrative examples which concern rectangular plate subjected to different load cases with various boundary conditions. The obtained results through this numerical investigation are compared with those obtained by the Finite Element Method (FEM) or the analytic formulas.

New Geometrical Model of Woven Fabric Taking into Account the Change of Its Form, Size and Lateral Bending

This paper suggests a technique for the construction of fabric three-dimensional models based on the piecewise continuous representation of the thread surface. The approach allows to exclude such simulation defects as “interpenetration” of threads in the fabric, and takes into account the geometric features of the thread structure, the possibility of their bending in several planes, and changes in the form and sectional area. The geometric models obtained can be used to analyse the forces and deformations occurring in the fabric with the help of the finite-element simulation programs in various specialised software complexes, such as ANSYS, LS-DYNA, ABAQUS, NASTRAN etc. The structure parameters which are difficult to calculate using standard techniques, e.g. the pore volume and contact area between threads can be calculated with the help of describing the fabric three-dimensional structure.

Efficient Convergence for a Higher-Order Unstructured Finite Volume Solver for Compressible Flows

This Paper presents a three-dimensional higher-order-accurate finite volume algorithm for the solution of steady-state compressible flow problems. Higher-order accuracy is achieved by constructing a piecewise continuous representation of the average solution values using the -exact reconstruction scheme. The pseudo-transient continuation method is employed to reduce the solution of the discretized system of nonlinear equations into the solution of a series of linear systems, which are subsequently solved using the generalized minimal residual (GMRES) method. This Paper considers several preconditioning methods in conjunction with different matrix reordering algorithms and shows that the proposed preconditioner based on inner GMRES iterations can enhance the convergence speed and reduce the memory cost of the solver. Moreover, when starting from a lower-order solution as the initial condition, this Paper shows that ramping up the Courant–Friedrichs–Lewy (CFL) number accelerates the convergence rate. Finally, this Paper verifies the developed finite volume algorithm by solving a set of test problems, in which optimal solution convergence with mesh refinement is attained.

Microfacet BSDFs Generated from NDFs and Explicit Microgeometry

Microfacet distributions are considered nowadays as a reference for physically plausible BSDF representations. Many authors have focused on their physical and mathematical correctness, while introducing means to enlarge the range of possible appearances. This article is dedicated to Normal Distribution Functions (NDFs) and the influence of their shape on the rendered material aspect. We provide a complete framework for studying the impact of NDFs on the observed Bidirectional Scattering Distribution Functions (BSDFs). To explore very general NDFs, manually controlled by the user, and including anisotropic materials, we propose to use a piecewise continuous representation. It is derived with its associated Smith shadowing-masking function and importance sampling formulations for ensuring efficient global illumination computations. A new procedure is also proposed in this article for generating an explicit geometric micro-surface, used to evaluate the validity of analytic models and multiple scattering effects. The results are produced with a computer-generated process using path tracing. They show that this generation procedure is suitable with any NDF model, independently from its shape complexity.

VOF evaluation of the surface tension by using variational representation and Galerkin interpolation projection

In this work we propose a variational approach with cell-to-point Galerkin projections for studying two-phase interface advection problems dominated by surface tension. A Volume Of Fluid (VOF) algorithm is used for tracking and locating the evolution of the two-phase interface on a Cartesian grid and a finite element numerical scheme for solving the velocity-pressure state. The velocity field that drives the evolution of this interface is computed from the weak form of the Navier-Stokes equation where the surface tension force is represented in variational form by the continuous surface force (CSF) and continuous surface stress (CSS) methods. Standard numerical approaches solve the strong form of the Navier-Stokes equations and define the CSS term by taking the divergence of the surface tension tensor. This computation of the divergence term results in a singular force which is difficult to compute when the grid is refined since the tensor is computed in a discontinuous cell-by-cell way. In this work we use the variational formulation of the Navier-Stokes equation and avoid differentiation. The tensor, which is a function of the unit normal, is evaluated over regular Sobolev spaces by using a cell-to-point Galerkin projection. This allows a regular piece-wise continuous representation of the surface tensor and the unit normal based on the VOF reconstruction. In standard approaches the CSF surface force is computed by using the curvature, which is the divergence of the unit normal. In this paper we recover the curvature with point-wise Galerkin projection avoiding direct differentiation. Tests on convergence for two and three-dimension in the static and dynamical cases are reported to show the correct representation in the desired spaces. This method is also natural for coupling non uniform grid computation of the fluid with Cartesian grid of the VOF algorithm.

A new surface tension VOF evaluation by using variational representation and Galerkin interpolation projection

In this work we propose a new algorithm for studying two-phase interface advection problems dominated by surface tension. We use a Volume Of Fluid (VOF) algorithm for studying the evolution of the two-phase interface on a Cartesian grid and a finite element numerical scheme for the velocity-pressure state. The velocity field that drives the evolution of this interface is obtained by solving the weak form of the Navier-Stokes equation where the surface tension force is not defined in a singular way. With standard numerical approaches that solve the strong form of the Navier-Stokes equations the surface force is determined by taking the divergence of the surface tension tensor. The computation of the divergence term results in a force which is non-convergent when the grid is refined since the tensor is computed in a discontinuous cell-by-cell way. In the past this approach was proposed with artificial different smoothing schemes in order to compute such a singular force. In this work we use the variational formulation of the Navier-Stokes equation and avoid differentiation. The tensor which is a function of the unit normal is evaluated by a Galerkin projection over regular Sobolev spaces. This allows the piece-wise continuous representation of the surface tensor and the unit normal based on the VOF reconstruction. Tests on convergence for two and three-dimension in the static and dynamical cases are reported to show the correct representation in the desired spaces. This method is also natural for coupling non uniform grid computation of the fluid with Cartesian grid of the VOF algorithm.

Roe solver with entropy corrector for uncertain hyperbolic systems

This paper deals with intrusive Galerkin projection methods with a Roe-type solver for treating uncertain hyperbolic systems using a finite volume discretization in physical space and a piecewise continuous representation at the stochastic level. The aim of this paper is to design a cost-effective adaptation of the deterministic Dubois and Mehlman corrector to avoid entropy-violating shocks in the presence of sonic points. The adaptation relies on an estimate of the eigenvalues and eigenvectors of the Galerkin Jacobian matrix of the deterministic system of the stochastic modes of the solution and on a correspondence between these approximate eigenvalues and eigenvectors for the intermediate states considered at the interface. We derive some indicators that can be used to decide where a correction is needed, thereby reducing the computational costs considerably. The effectiveness of the proposed corrector is assessed on the Burgers and Euler equations including sonic points.

A spline-based algorithm for continuous time-delay estimation using sampled data

Time delay estimation (TDE) lies at the heart of signal processing algorithms in a broad range of application areas, including communications, coherent imaging, speech processing, and acoustics. In medical ultrasound for example, TDE is used in blood flow estimation, tissue motion measurement, tissue elasticity estimation, phase aberration correction, and a number of other algorithms. Because of its central significance, TDE accuracy, precision, and computational cost are of critical importance. Furthermore, because TDE is typically performed on sampled signals-and delay estimates are usually desired over a continuous domain-time delay estimator performance should be considered in conjunction with associated interpolation. In this paper we present a new time-delay estimator that directly determines continuous time-delay estimates from sampled data. The technique forms a spline-based, piecewise continuous representation of the reference signal then solves for the minimum of the sum squared error between the reference and the delayed signals to determine their relative time delay. Computer simulation results clearly show that the proposed algorithm significantly outperforms other algorithms in terms of jitter and bias over a broad range of conditions. We also describe a modified version of the algorithm that includes companding with only a minor increase in computational cost.

High Schmidt number scalars in turbulence: Structure functions and Lagrangian theory

We demonstrate the existence of Batchelor’s viscous-convective subrange using direct numerical simulation (DNS) results to confirm the logarithmic dependence of the scalar structure function on the separation for the scalar field generated by stationary isotropic turbulence acting on a uniform mean scalar gradient. From these data we estimate the Batchelor constant B̃θ≈5. By integrating a piecewise continuous representation of the scalar variance spectrum we calculate the steady-state scalar variance as a function of Reynolds number and Schmidt number. Comparison with DNS results confirms the Reλ−1 behavior predicted from the spectral integration, but with a coefficient about 60% too small. In the large Reynolds number limit the data give a value of 2.5 for the mechanical-to-scalar time scale ratio. The dependence of the data for the scalar variance on Schmidt number agrees very well with the spectral integration using the values of the Batchelor constant estimated from the structure function. We also carry out an exact Lagrangian analysis of the scalar variance and structure function, explicitly relating the Batchelor constant to the Lyapunov exponent for the separation of pairs of fluid particles within the turbulence dissipation subrange. Our results, particularly for the scalar variance, illustrate explicitly the singular nature of the zero diffusivity limit. For finite values of the Schmidt number and Reynolds number the viscous-convective subrange contribution to the variance can be significant even at moderate values of the Reynolds number.

Reply by Authors to J. Smolderen

fact that the derivative of density with respect to velocity, which occurs in the function g of Eq. (1), is not correctly represented. They accept this incorrect representation on the grounds that this derivative does not occur explicitly in the basic physical equations of the problem, and probably mean, thereby, the integral version of these equations. They conelude from this, that a wildly oscillating representation of dpldVc may be used, as long as its integral, p, approximates the correct p(Vc). This could probably be justified by consideration of the integral form of the equations of motion, and we will accept it, although mathematicians might object to violent tinkering with coefficients of partial differential equations. However, the application that Cahn and Garcia make of this principle is completely erroneous. In fact, they push things too far when they consider a piecewise continuous representation of density in function of velocity as shown on their Fig. 3, which leads to infinite values for the density derivative at the points of discontinuity. This is an essential fact which the authors appear to have missed. If one considers the piecewise continuous representation of Fig. 3 as the limit of a smooth curve with very large slope in small regions around the points of discontinuity , one is led to conclude from the second Eq. (1), that 3 c/dVc will be of a similar large order of magnitude. In the limiting case of a discontinuous density, one should therefore expect finite jumps in the potential. These jumps are not considered by the authors and their justification for the use of the transformation (7) is therefore not valid, as expected from our earlier arguments. In the last paragraph of their section Outline of Theory, Cahn and Garcia present another justification for their assumption about the validity of replacing the actual density by an approximation with wildly oscillating derivative. Although we do not reject this assumption and have shown that the error lies elsewhere, we cannot accept this justification, based on the consideration of the electric analogy tank with variable depth. The equation solved by the analogy is

Complex Modulus of a Cohesive Soil from Stress Relaxation Response using the One-Sided Fourier Transform

The storage and dissipation resistance to deformation of a cohesive soil as represented by the complex modulus is obtained from stress relaxation response using the one‐sided Fourier transform. The soil response is nonlinear. An approximate method of evaluating the transform with digital computer methods is based on the use of straight line segments to form a piecewise continuous representation of the relaxation response. This allows comparison of static and dynamic response characteristics as functions of applied strain level and moisture content through the transformation from the time domain to the frequency domain. For each strain level, large and small time plateau regions of complex modulus are connected with a transition region showing an increase in modulus with increase in frequency. Complex modulus decreases with increased strain level and increased moisture content.

Complex Modulus of a Cohesive Soil from Stress Relaxation Response using the One-Sided Fourier Transform

The storage and dissipation resistance to deformation of a cohesive soil as represented by the complex modulus is obtained from stress relaxation response using the one-sided Fourier transform. The soil response is nonlinear. An approximate method of evaluating the transform with digital computer methods is based on the use of straight line segments to form a piecewise continuous representation of the relaxation response. This allows comparison of static and dynamic response characteristics as functions of applied strain level and moisture content through the transformation from the time domain to the frequency domain. For each strain level, large and small time plateau regions of complex modulus are connected with a transition region showing an increase in modulus with increase in frequency. Complex modulus decreases with increased strain level and increased moisture content.