Highly accurate numerical solutions with repeated Richardson extrapolation for 2D laplace equation

A theoretical basis is presented for the repeated Richardson extrapolation (RRE) to reduce and estimate the discretization error of numerical solutions for heat conduction. An example application is described for the 2D Laplace equation using the finite difference method, a domain discretized with uniform grids, second-order accurate approximations, several variables of interest, Dirichlet boundary conditions, grids with up to 8,193 × 8,193 nodes, a multigrid method, single, double and quadruple precisions and up to twelve Richardson extrapolations. It was found that: (1) RRE significantly reduces the discretization error (for example, from 2.25E-07 to 3.19E-32 with nine extrapolations and a 1,025 × 1,025 grid, yielding an order of accuracy of 19.1); (2) the Richardson error estimator works for numerical results obtained with RRE; (3) a higher reduction of the discretization error with RRE is achieved by using higher calculation precision, a larger number of extrapolations, a larger number of grids and correct error orders; and (4) to obtain a given value error, much less CPU time and RAM memory are required for the solution with RRE than without it.     

Verified predictions of shape sensitivities in wall-bounded turbulent flows by an adaptive finite-element method

A Continuous Sensitivity Equation (CSE) method is presented for shape parameters in turbulent wall-bounded flows modeled with the standard k–? turbulence model with wall functions. Differentiation of boundary conditions and their complex dependencies on shape parameters, including the two-velocity scale wall functions, is presented in details along with the appropriate methodology required for the CSE method. To ensure accuracy, grid convergence and to reduce computational time, an adaptive finite-element method driven by asymptotically exact error estimations is used. The adaptive process is controlled by error estimates on both flow and sensitivity solutions. Firstly, the proposed approach is applied on a problem with a closed-form solution, derived using the Method of the Manufactured Solution to perform Code Verification. Results from adaptive grid refinement studies show Verification of flow and sensitivity solvers, error estimators and the adaptive strategy. Secondly, we consider turbulent flows around a square cross-section cylinder in proximity of a solid wall. We examine the quality of the numerical solutions by performing Solution Verification and Validation. Then, Sensitivity Analysis of these turbulent flows is performed to investigate the ability of the method to deal with non-trivial geometrical changes. Sensitivity information is used to estimate uncertainties in the flow solution caused by uncertainties in the shape parameter and to perform fast evaluation of flows on nearby configurations.     

An adaptive heuristic algorithm for VLSI test vectors selection

The increasing complexity of today’s system-on-a-chip designs is putting more pressure on the already stressed design verification process. The verification plan must cover several individual cores as well as the overall chip design. Conditions to be verified are identified by the system’s architects, the designers, and the verification team. Testing for these conditions is a must for the design to tape out, especially for high priority conditions. A significant bottleneck in the verification process of such designs is that not enough time is usually given to the final coverage phase, which makes computing cycles very precious. Thus, intelligent selection of test vectors that achieve the best coverage using the minimum number of computing cycles is crucial for on time tape out. This paper presents a novel heuristic algorithm for test vectors selection. The algorithm attempts to achieve the best coverage level while minimizing the required number of computing cycles.     

Verifying exactness of relaxations for robust semi-definite programs by solving polynomial systems

In this paper, robust semi-definite programs are considered with the goal of verifying whether a particular LMI relaxation is exact. A procedure is presented showing that verifying exactness amounts to solving a polynomial system. The main contribution of the paper is a new algorithm to compute all isolated solutions of a system of polynomials. Standard techniques in computational algebra, often referred to as Stetter’s method [H.J. Stetter, Numerical Polynomial Algebra, SIAM, 2004], involve the computation of a Gröbner basis of the ideal generated by the polynomials and further require joint eigenvector computations in order to arrive at the zeros of the polynomial system. Our algorithm does neither require structural knowledge on the polynomial system, nor does it rely on the computation of joint eigenvectors.     

Verification of numerical simulation method for entropy generation of radiation heat transfer in semitransparent medium

The numerical simulation method of radiative entropy generation in participating media presented by Caldas and Semiao [Entropy generation through radiative transfer in participating media: analysis and numerical computation. JQSRT 2005;96:423-37] is extended to analyze the radiative entropy generation in the enclosures filled with semitransparent media. A discrete ordinates method is used to solve radiative transfer equation and radiative entropy generation. Two different examples are employed to verify the numerical simulation method of radiative entropy generation in the enclosure. Numerical results of dimensionless radiative entropy generation of enclosure are identical to that of entire thermodynamics analysis for the enclosure system. This numerical simulation method can be used in the entropy generation analysis of high-temperature systems such as boilers and furnaces, in which radiation is the dominant mode of heat transfer.     

Upper error bounds on calculated outputs of interest for linear and nonlinear structural problems

This Note introduces new strict upper error bounds on outputs of interest for linear as well as time-dependent nonlinear structural problems calculated by the finite element method. Small-displacement problems without softening, such as (visco)plasticity problems, are included through the standard thermodynamics framework involving internal state variables. To cite this article: P. Ladevèze, C. R. Mecanique 334 (2006).     

Comparison of refinement criteria for structured adaptive mesh refinement

We have investigated and analyzed the grid convergence issues for an adaptive mesh refinement (AMR) code. We have found that the numerical results for the AMR grid may have a larger error than those for the unrefined uniform grid. After a detailed analysis, we have found that the numerical solution at the coarse-fine interface between different levels of the grid converges only in the first-order accuracy. Therefore, the error near the coarse-fine interface can quickly dominate the error in the other regions if the coarse-fine interface is active and not covered by the fine grid. We propose, implement, and compare several refinement criteria. Some of them can catch the large-error region near the coarse-fine interface and refine them with the fine grid.     

Students’ use of technological tools for verification purposes in geometry problem solving

Despite its importance in mathematical problem solving, verification receives rather little attention by the students in classrooms, especially at the primary school level. Under the hypotheses that (a) non-standard tasks create a feeling of uncertainty that stimulates the students to proceed to verification processes and (b) computational environments - by providing more available tools compared to the traditional environment - might offer opportunities for more frequent usage of verification techniques, we posed to 5th and 6th graders non-routine problems dealing with area of plane irregular figures. The data collected gave us evidence that computational environments allow the development of verification processes in a wider variety compared to the traditional paper-and-pencil environment and at the same time we had the chance to propose a preliminary categorization of the students’ verification processes under certain conditions.     

Analysis of meeting protocols by formalisation, simulation, and verification

Organizations depend on regular meetings to carry out their everyday tasks. When carried out successfully, meetings offer a common medium for participants to exchange ideas and make decisions. However, many meetings suffer from unfocused discussions or irrelevant dialogues. To study meetings in detail, we first formalize general properties of meetings and a generic meeting protocol to specify how roles in a meeting should interact to realize these properties. This generic protocol is used as a starting point to study real-life meetings. Next, an example meeting is simulated using the generic meeting protocol. The general properties are formally verified in the simulation trace. Next, these properties are also verified formally against empirical data of a real meeting in the same context. A comparison of the two traces reveals that a real meeting is more robust since when exceptions happen and the rules of the protocol are violated, these exceptions are handled effectively. Given this observation, a more refined protocol is specified that includes exception-handling strategies. Based on this refined protocol a meeting is simulated that closely resembles the real meeting. This protocol is then validated against another set of data from another real meeting. By iteratively adding exception handling rules, the protocol is enhanced to handle a variety of situations successfully.