Chandra Shekhar Upadhyay

Bio: Chandra Shekhar Upadhyay is an academic researcher from Indian Institute of Technology Kanpur. The author has contributed to research in topics: Finite element method & Spectral element method. The author has an hindex of 19, co-authored 76 publications receiving 1476 citations. Previous affiliations of Chandra Shekhar Upadhyay include Texas A&M University & Indian Institute of Technology, Jodhpur.

Validation of A-Posteriori Error Estimators by Numerical Approach

Abstract: : A numerical methodology which determines the quality (or robustness) of a-posteriori error estimators is described. The methodology accounts precisely for the factors which affect the quality of error estimators for finite-element solutions of linear elliptic problems, namely, the local geometry of the grid and the structure of the solution. The methodology can be employed to check the robustness of any estimator for the complex grids which are used in engineering computations.

A model study of the quality of a posteriori error estimators for linear elliptic problems. Error estimation in the interior of patchwise uniform grids of triangles

Abstract: This paper is the first in a series in which we discuss computational methodologies for checking the quality of a posteriori error estimators for finite element approximations of linear elliptic problems. In this first part we study the asymptotic properties of error estimators in the interior of patchwise uniform grids of triangles. A completely numerical methodology for the analysis of the quality of estimators is presented. Results from the application of the methodology to the study of the quality of several well-known error estimators are reported. In subsequent papers we shall discuss methods to study the properties of estimators for meshes of quadrilaterals, non-uniform grids, at boundaries, grid-interfaces and near-singular points.

A posteriori estimation and adaptive control of the pollution error in the h‐version of the finite element method

Abstract: : We studied the pollution-error in the h-version of the finite element method and its effect on the quality of the local error indicators (resp. the quality of the derivatives recovered by local postprocessing) in the interior of the mesh. Here we show that it is possible to construct a-posteriori estimates of the pollution-error in a patch of elements by employing the local error indicators over the entire mesh. We also give an adaptive algorithm for the local control of the pollution-error in a patch of elements of interest.

Pollution Error in the h-Version of the Finite Element Method and the Local Quality of the Recovered Derivatives.

Abstract: In this paper, we address the quality of the solution derivatives which are recovered from finite element solutions by local averaging schemes. As an example, we consider the Zienkiewicz-Zhu superconvergent patch-recovery scheme (the ZZ-SPR scheme), and we study its accuracy in the interior of the mesh for finite element approximations of solutions of Laplace's equation in polygonal domains. We will demonstrate the following: (1) In general, the accuracy of the derivatives recovered by the ZZ-SPR or any other local averaging scheme may not be higher than the accuracy of the derivatives computed directly from the finite element solution. (2) If the mesh is globally adaptive (i.e. it is nearly equilibrated in the energy-norm) then we can, practically always, gain in accuracy by employing the recovered derivatives instead of the derivatives computed directly from the finite element solution. (3) It is possible to guarantee that the recovered solution-derivatives have higher accuracy than the derivatives computed directly from the finite element solution, in any patch of elements of interest, by employing a mesh which is adaptive only with respect to the patch of interest (i.e. it is nearly equilibrated in a weighted energy-norm). (4) In practice, we are often interested in obtaining highly accurate derivatives (or heat-fluxes, stresses, etc.) only in a few critical regions which are identified by a preliminary analysis. A grid which is adaptive only with respect to the critical regions of interest may be much more economical for this purpose because it may achieve the desired accuracy by employing substantially fewer degrees of freedom than a globally adaptive grid which achieves comparable accuracy in the critical regions.

A Posteriori Error Estimation in Finite Element Analysis

Abstract: This monograph presents a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics. The study primarily focuses on methods for linear elliptic boundary value problems. However, error estimation for unsymmetrical systems, nonlinear problems, including the Navier-Stokes equations, and indefinite problems, such as represented by the Stokes problem are included. The main thrust is to obtain error estimators for the error measured in the energy norm, but techniques for other norms are also discussed.

Quantification of uncertainty in computational fluid dynamics

Abstract: This review covers Verification, Validation, Confirmation and related subjects for computational fluid dynamics (CFD), including error taxonomies, error estimation and banding, convergence rates, surrogate estimators, nonlinear dynamics, and error estimation for grid adaptation vs Quantification of Uncertainty.

Verification and Validation in Scientific Computing

Abstract: Advances in scientific computing have made modelling and simulation an important part of the decision-making process in engineering, science, and public policy. This book provides a comprehensive and systematic development of the basic concepts, principles, and procedures for verification and validation of models and simulations. The emphasis is placed on models that are described by partial differential and integral equations and the simulations that result from their numerical solution. The methods described can be applied to a wide range of technical fields, from the physical sciences, engineering and technology and industry, through to environmental regulations and safety, product and plant safety, financial investing, and governmental regulations. This book will be genuinely welcomed by researchers, practitioners, and decision makers in a broad range of fields, who seek to improve the credibility and reliability of simulation results. It will also be appropriate either for university courses or for independent study.

Verification and Validation in Computational Fluid Dynamics

Abstract: Verification and validation (V&V) are the primary means to assess accuracy and reliability in computational simulations. This paper presents an extensive review of the literature in V&V in computational fluid dynamics (CFD), discusses methods and procedures for assessing V&V, and develops a number of extensions to existing ideas. The review of the development of V&V terminology and methodology points out the contributions from members of the operations research, statistics, and CFD communities. Fundamental issues in V&V are addressed, such as code verification versus solution verification, model validation versus solution validation, the distinction between error and uncertainty, conceptual sources of error and uncertainty, and the relationship between validation and prediction. The fundamental strategy of verification is the identification and quantification of errors in the computational model and its solution. In verification activities, the accuracy of a computational solution is primarily measured relative to two types of highly accurate solutions: analytical solutions and highly accurate numerical solutions. Methods for determining the accuracy of numerical solutions are presented and the importance of software testing during verification activities is emphasized. The fundamental strategy of validation is to assess how accurately the computational results compare with the experimental data, with quantified error and uncertainty estimates for both. This strategy employs a hierarchical methodology that segregates and simplifies the physical and coupling phenomena involved in the complex engineering system of interest. A hypersonic cruise missile is used as an example of how this hierarchical structure is formulated. The discussion of validation assessment also encompasses a number of other important topics. A set of guidelines is proposed for designing and conducting validation experiments, supported by an explanation of how validation experiments are different from traditional experiments and testing. A description is given of a relatively new procedure for estimating experimental uncertainty that has proven more effective at estimating random and correlated bias errors in wind-tunnel experiments than traditional methods. Consistent with the authors’ contention that nondeterministic simulations are needed in many validation comparisons, a three-step statistical approach is offered for incorporating experimental uncertainties into the computational analysis. The discussion of validation assessment ends with the topic of validation metrics, where two sample problems are used to demonstrate how such metrics should be constructed. In the spirit of advancing the state of the art in V&V, the paper concludes with recommendations of topics for future research and with suggestions for needed changes in the implementation of V&V in production and commercial software.

The stochastic finite element method: Past, present and future

Abstract: A powerful tool in computational stochastic mechanics is the stochastic finite element method (SFEM). SFEM is an extension of the classical deterministic FE approach to the stochastic framework i.e. to the solution of static and dynamic problems with stochastic mechanical, geometric and/or loading properties. The considerable attention that SFEM received over the last decade can be mainly attributed to the spectacular growth of computing power rendering possible the efficient treatment of large-scale problems. This article aims at providing a state-of-the-art review of past and recent developments in the SFEM area and indicating future directions as well as some open issues to be examined by the computational mechanics community in the future.