Space Discretization Error Research Articles

The edge-based smoothed FEM with [formula omitted]-Bathe implicit temporal discretization scheme for the analyses of underwater wave propagation problems

The underwater acoustic wave propagation problem is an important issue in ocean engineering field. The classical finite element approach with edge-based smoothed gradient smoothing technique is incorporated into the novel ρ∞-Bathe implicit direct time integration scheme in this paper for solving underwater transient acoustic wave propagation problems. This edge-based smoothed FEM (ES-FEM) and ρ∞-Bathe method are employed to perform the space and time domain discretization, respectively. Due to the appropriate softening effects from the ES-FEM, the space discretization errors can be markedly reduced. Meanwhile, the possible numerical error related to the temporal discretization also can be significantly suppressed by the novel ρ∞-Bathe method which possesses the controllable numerical damping effects. Through detailed analysis of the total dispersion error, we find that the numerical results from the proposed scheme can converge monotonically to the exact solutions when employing the decreasing temporal discretization intervals and the possible numerical anisotropy effects are also effectively suppressed. These excellent numerical performance clearly distinguish the present method from the conventional approaches and make it to be particularly suitable for complicated acoustic wave propagation analysis. Finally, various typical numerical experiments are performed to illustrate the capacities of the proposed scheme in tackling underwater transient acoustic wave propagation problems.

A high order convergent numerical method for singularly perturbed time dependent problems using mesh equidistribution

The purpose of this paper is to introduce a high order convergent numerical method for singularly perturbed time dependent problems using mesh equidistribution. The discretization is based on the backward Euler scheme in time and a high order non-monotone scheme in space. In time direction we consider a uniform mesh, while in spatial direction we construct an adaptive mesh through equidistribution of a monitor function involving appropriate power of the solution’s second derivative. The method is analysed in two steps, splitting the time and space discretization errors. We establish that the method is uniformly convergent with optimal order having order one in time and order four in space. Further, we use the Richardson extrapolation technique for improving the order of convergence from one to two in time. Numerical experiments are presented to confirm the theoretically proven convergence result.

Numerical Errors in Unsteady Flow Simulations

This article discusses numerical errors in unsteady flow simulations, which may include round-off, statistical, iterative, and time and space discretization errors. The estimation of iterative and discretization errors and the influence of the initial condition on unsteady flows that become periodic are discussed. In this latter case, the goal is to determine the simulation time required to reduce the influence of the initial condition to negligible levels. Two one-dimensional, unsteady manufactured solutions are used to illustrate the interference between the different types of numerical errors. One solution is periodic and the other includes a transient region before it reaches a steady-state. The results show that for a selected grid and time-step, statistical convergence of the periodic solution may be achieved at significant lower error levels than those of iterative and discretization errors. However, statistical convergence deteriorates when iterative convergence criteria become less demanding, grids are refined, and Courant number increased.For statistically converged solutions of the periodic flow and for the transient solution, iterative convergence criteria required to obtain a negligible influence of the iterative error when compared to the discretization error are more strict than typical values found in the open literature. More demanding criteria are required when the grid is refined and/or the Courant number is increased. When the numerical error is dominated by the iterative error, it is pointless to refine the grid and/or reduce the time-step. For solutions with a numerical error dominated by the discretization error, three different techniques are applied to illustrate how the discretization uncertainty can be estimated, using grid/time refinement studies: three data points at a fixed Courant number; five data points involving three time steps for the same grid and three grids for the same time-step; five data points including at least two grids and two time steps. The latter two techniques distinguish between space and time convergence, whereas the first one combines the effect of the two discretization errors.

Quantifying Discretization Errors in Electrophoretically-Guided Micro Additive Manufacturing.

This paper presents process models for a new micro additive manufacturing process termed Electrophoretically-guided Micro Additive Manufacturing (EPμAM). In EPμAM, a planar microelectrode array generates the electric potential distributions which cause colloidal particles to agglomerate and deposit in desired regions. The discrete microelectrode array nature and the used pulse width modulation (PWM) technique for microelectrode actuation create unavoidable process errors—space and time discretization errors—that distort particle trajectories. To combat this, we developed finite element method (FEM) models to study trajectory deviations due to these errors. Mean square displacement (MSD) analysis of the computed particle trajectories is used to compare these deviations for several electrode geometries. The two top-performing electrode geometries evaluated by MSD were additionally investigated through separate case studies via geometry variation and MSD recomputation. Furthermore, separate time-discretization error simulations are also studied where electrode actuating waveforms were simulated. The mechanical impulse of the electromechanical force, generated from these waveforms is used as the basis for comparison. The obtained results show a moderate MSDs variability and significant differences in the computed mechanical impulses for the actuating waveforms. The observed limitations of the developed process model and of the error comparison technique are briefly discussed and future steps are recommended.

Rotation Hough Transform

This article discusses how image space discretization errors affect the standard Hough transform (SHT). It is observed that the discretization errors are different on various columns, that is, they are related to the θ values. Furthermore, the columns corresponding to θ=0° and θ=90° are seen to have the smallest errors. Based on this observation, a new Hough transform (rotation Hough transform, RoHT) is proposed. The given image is rotated by Δθ and the SHT is employed, generating the HT data on the columns corresponding to θ=0° and θ=90° only. After the image is rotated 90°/Δθ times, the HT columns generated are registered to the HT matrix of the original image. In this way, only θ=0° and θ=90° are used to calculate the votes for each cell in HT space, and the errors are minimized. The experiments verified the performance of the proposed method.

An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow

In this paper we derive an a posteriori error estimate for the numerical approximation of the solution of a system modeling the flow of two incompressible and immiscible fluids in a porous medium. We take into account the capillary pressure, which leads to a coupled system of two equations: parabolic and elliptic. The parabolic equation may become degenerate, i.e., the nonlinear diffusion coefficient may vanish over regions that are not known a priori. We first show that, under appropriate assumptions, the energy-type norm differences between the exact and the approximate nonwetting phase saturations, the global pressures, and the Kirchhoff transforms of the nonwetting phase saturations can be bounded by the dual norm of the residuals. We then bound the dual norm of the residuals by fully computable a posteriori estimators. Our analysis covers a large class of conforming, vertex-centered finite volume-type discretizations with fully implicit time stepping. As an example, we focus here on two approaches: a ``mathematical'' scheme derived from the weak formulation, and a phase-by-phase upstream weighting ``engineering'' scheme. Finally, we show how the different error components, namely the space discretization error, the time discretization error, the linearization error, the algebraic solver error, and the quadrature error can be distinguished and used for making the calculations efficient.

Accurate finite element modeling of linear elastodynamics problems with the reduced dispersion error

It is known that the reduction in the finite element space discretization error for elastodynamics problems is related to the reduction in numerical dispersion of finite elements. In the paper, we extend the modified integration rule technique for the mass and stiffness matrices to the dispersion reduction of linear finite elements for linear elastodynamics. The analytical study of numerical dispersion for the modified integration rule technique and for the averaged mass matrix technique is carried out in the 1-D, 2-D and 3-D cases for harmonic plane waves. In the general case of loading, the numerical study of the effectiveness of the dispersion reduction techniques includes the filtering technique (developed in our previous papers) that identifies and removes spurious high-frequency oscillations. 1-D, 2-D and 3-D impact problems for which all frequencies of the semi-discrete system are excited are solved with the standard approach and with the new dispersion reduction technique. Numerical results show that compared with the standard mass and stiffness matrices, the simple dispersion reduction techniques lead to a considerable decrease in the number of degrees of freedom and computation time at the same accuracy, especially for multi-dimensional problems. A simple quantitative estimation of the effectiveness of the finite element formulations with reduced numerical dispersion compared with the formulation based on the standard mass and stiffness matrices is suggested.

Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs

Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D⊂ℝ d are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L 2(D)-orthogonal bases, and on viewing the coefficients of these expansions as random parameters y=y(ω)=(y i (ω)). This yields an equivalent parametric deterministic PDE whose solution u(x,y) is a function of both the space variable x∈D and the in general countably many parameters y. We establish new regularity theorems describing the smoothness properties of the solution u as a map from y∈U=(−1,1)∞ to $V=H^{1}_{0}(D)$ . These results lead to analytic estimates on the V norms of the coefficients (which are functions of x) in a so-called “generalized polynomial chaos” (gpc) expansion of u. Convergence estimates of approximations of u by best N-term truncated V valued polynomials in the variable y∈U are established. These estimates are of the form N −r , where the rate of convergence r depends only on the decay of the random input expansion. It is shown that r exceeds the benchmark rate 1/2 afforded by Monte Carlo simulations with N “samples” (i.e., deterministic solves) under mild smoothness conditions on the random diffusion coefficients. A class of fully discrete approximations is obtained by Galerkin approximation from a hierarchic family $\{V_{l}\}_{l=0}^{\infty}\subset V$ of finite element spaces in D of the coefficients in the N-term truncated gpc expansions of u(x,y). In contrast to previous works, the level l of spatial resolution is adapted to the gpc coefficient. New regularity theorems describing the smoothness properties of the solution u as a map from y∈U=(−1,1)∞ to a smoothness space W⊂V are established leading to analytic estimates on the W norms of the gpc coefficients and on their space discretization error. The space W coincides with $H^{2}(D)\cap H^{1}_{0}(D)$ in the case where D is a smooth or convex domain. Our analysis shows that in realistic settings a convergence rate $N_{\mathrm{dof}}^{-s}$ in terms of the total number of degrees of freedom N dof can be obtained. Here the rate s is determined by both the best N-term approximation rate r and the approximation order of the space discretization in D.

Numerical investigation of effectivity indices of space-time error indicators for Navier–Stokes equations

In this paper we propose two error indicators aimed at estimating the space discretization error and the time discretization error for the unsteady Navier–Stokes equations. We define a space error indicator for evaluating the quality of the mesh and a time error indicator for evaluating the time discretization error. Moreover, we verify the reliability of the estimations through numerical experiments and we propose an effective space-time adaptive strategy for the unsteady Navier–Stokes equations. Such technique is based on two residual-based error indicators that suitably drive the mesh and the timestep-length modifications. Adaptive simulations show that the presented strategy allows to obtain accurate solutions in efficient way.

Convergence rates of best N-term Galerkin approximations for a class of elliptic sPDEs

Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D⊂ℝd are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L 2(D)-orthogonal bases, and on viewing the coefficients of these expansions as random parameters y=y(ω)=(y i (ω)). This yields an equivalent parametric deterministic PDE whose solution u(x,y) is a function of both the space variable x∈D and the in general countably many parameters y. We establish new regularity theorems describing the smoothness properties of the solution u as a map from y∈U=(−1,1)∞ to $V=H^{1}_{0}(D)$. These results lead to analytic estimates on the V norms of the coefficients (which are functions of x) in a so-called “generalized polynomial chaos” (gpc) expansion of u. Convergence estimates of approximations of u by best N-term truncated V valued polynomials in the variable y∈U are established. These estimates are of the form N −r , where the rate of convergence r depends only on the decay of the random input expansion. It is shown that r exceeds the benchmark rate 1/2 afforded by Monte Carlo simulations with N “samples” (i.e., deterministic solves) under mild smoothness conditions on the random diffusion coefficients. A class of fully discrete approximations is obtained by Galerkin approximation from a hierarchic family $\{V_{l}\}_{l=0}^{\infty}\subset V$of finite element spaces in D of the coefficients in the N-term truncated gpc expansions of u(x,y). In contrast to previous works, the level l of spatial resolution is adapted to the gpc coefficient. New regularity theorems describing the smoothness properties of the solution u as a map from y∈U=(−1,1)∞ to a smoothness space W⊂V are established leading to analytic estimates on the W norms of the gpc coefficients and on their space discretization error. The space W coincides with $H^{2}(D)\cap H^{1}_{0}(D)$in the case where D is a smooth or convex domain. Our analysis shows that in realistic settings a convergence rate $N_{\mathrm{dof}}^{-s}$in terms of the total number of degrees of freedom N dof can be obtained. Here the rate s is determined by both the best N-term approximation rate r and the approximation order of the space discretization in D.

A posteriori error estimates by recovered gradients in parabolic finite element equations

This paper considers a posteriori error estimates by averaged gradients in second order parabolic problems. Fully discrete schemes are treated. The theory from the elliptic case as to when such estimates are asymptotically exact, on an element, is carried over to the error on an element at a given time. The basic principle is that the elliptic theory can be extended to the parabolic problems provided the time-step error is smaller than the space-discretization error. Numerical illustrations confirming the theoretical results are given. Our results are not practical in the sense that various constants can not be estimated realistically. They are conceptual in nature.

Adaptivity with Dynamic Meshes for Space-Time Finite Element Discretizations of Parabolic Equations

In this paper, we develop an error estimator and an adaptive algorithm for efficient solution of parabolic partial differential equations. The error estimator assesses the discretization error with respect to a given quantity of physical interest and separates the influence of the time and space discretizations. This allows us to set up an efficient adaptive strategy producing economical (locally) refined meshes for each time step and an adapted time discretization. The space and time discretization errors are equilibrated, leading to an efficient method.

Space Discretization Error of Methane Combustion Simulations in Turbulent Flow

Numerical investigation of methane combustion in a pipe with turbulent flow is studied. The space discretization error is investigated quantitatively and qualitatively, using the Richardson extrapolation and profiles comparisons. Comparison of the profiles indicates that the solution converges to a grid-independent solution. The Richardson method gives unsatisfactory results to determine the grid error, because of the rigidity of the method. A second-order polynomial is used as an alternative to the Richardson method. The results are more stable and have a better goodness of fit. The results of the simulations are compared with those of a similar experiment and the corresponding analytical solution.

Reliable Shape Optimization of Structures Subjected to Transient Dynamic Loading Using Genetic Algorithms

This paper presents a reliable method of solution of two dimensional shape optimization problems subjected to transient dynamic loads using Genetic Algorithms. Boundary curves undergoing shape changes have been represented by B‐splines. Automatic mesh generation and adaptive finite element analysis modules are integrated with Genetic algorithm code to carry out the shape optimization. Both space and time discretization errors are evaluated and appropriate finite element mesh and time step values as obtained iteratively are adopted for accurate dynamic response. Two demonstration problems have been solved, which show convergence to the optimal solution with number of generations. The boundary curve undergoing shape optimization shows smooth shape changes. The combinations of automatic mesh generator with proper boundary definition capabilities, analysis tool with error estimation and Genetic algorithm as optimization engine have been observed to behave as a satisfactory shape optimization environment to deal with real engineering problems.

Estimation of Numerical Errors Due to Time and Space Discretizations

In this paper, we propose a method to estimate time and space discretization errors of a three-dimensional magnetostatic model coupled with electric circuit equations. The estimator is based on the nonverification of the constitutive relationship which requires the calculation of admissible fields. The calculation on the whole time domain is described for the A-formulation. This estimator is tested on a transformer at no load and different schemes for time discretization are studied and compared.

Adaptive finite element analysis of structures subjected to transient dynamic loads using time marching scheme

Presents a method of obtaining accurate response using the finite element analysis for linear elastodynamic problems under transient dynamic loading. The accuracy obtained would depend on the discretization of both space and time. An integrated method is presented where both space and time discretization errors are evaluated and iteratively converges to a solution of desired accuracy through adaptive refinement. In this paper, it is proposed to solve the response using Newmark-β method. A numbers of examples are solved to demonstrate the effectiveness of the procedure.

Least-Squares Galerkin Methods for Parabolic Problems II: The Fully Discrete Case and Adaptive Algorithms

In this second part of our series on least-squares Galerkin methods for parabolic initial-boundary value problems we study the full discretization in time and space. These methods are based on the minimization of a least-squares functional for an equivalent first-order system over space and time with respect to suitable discrete spaces. Based on the analysis of the semi-discretization in time carried out in the first part, we construct a posteriori error estimators for the approximation error components associated with time and space. For the space discretization error we use a hierarchical basis estimator with respect to the functional minimization problem. We prove a strengthened Cauchy--Schwarz inequality between coarse and hierarchical surplus space and a bound on the condition number of the auxiliary problem, both uniform in the mesh-size h and the time-step $\tau$ implying efficiency and reliability of the error estimator. This allows for an adaptive strategy for both the time-step size and the spatial triangulation, keeping a proper balance between these two components. Numerical experiments illustrate the performance of the resulting adaptive schemes in time and space.

A posteriori error estimation and adaptivity for degenerate parabolic problems

Two explicit error representation formulas are derived for degenerate parabolic PDEs, which are based on evaluating a parabolic residual in negative norms. The resulting upper bounds are valid for any numerical method, and rely on regularity properties of solutions of a dual parabolic problem in nondivergence form with vanishing diffusion coefficient. They are applied to a practical space-time discretization consisting of C 0 piecewise linear finite elements over highly graded unstructured meshes, and backward finite differences with varying time-steps. Two rigorous a posteriori error estimates are derived for this scheme, and used in designing an efficient adaptive algorithm, which equidistributes space and time discretization errors via refinement/coarsening. A simulation finally compares the behavior of the rigorous a posteriori error estimators with a heuristic approach, and hints at the potentials and reliability of the proposed method.

Evaluation of the indoor temperature field using a given air velocity distribution

A new method for compensating the space discretization error introduced when the fixed flow field is considered for the dynamic models of temperature distribution is presented. It is proved that the method generally used in literature is a particular solution of the proposed one. Moreover, it results in a continuous-time model, for which the integrating method becomes a free choice and a state-space representation is possible. The numerical model was experimentally validated, the comparison, both in the time and in the frequency domains, between simulation and measured results showing good agreement. The presented dynamic model increases the calculation speed and it can be analysed with the tools developed in control theory.

A fully adaptive MOL-treatment of parabolic 1-D problems with extrapolation techniques

A fully adaptive method is presented for the numerical solution of highly nonlinear, coupled systems of parabolic differential equations in one space dimension. Time discretization is by means of the linearly implicit Euler discretization. Space discretization is by finite differences on nonuniform grids. Both basic discretizations are combined with extrapolation. Based on local error estimates for both the time and the space discretization error, the accuracy of the numerical approximation is controlled and the discretization stepsizes are adapted automatically and simultaneously. The algorithm is implemented in a user friendly software package, PDEX1M.