arXiv subjects

K. B. Nakshatrala

Publications and source records attributed to K. B. Nakshatrala.

At least 55 records · Page 3Linked to original sources

arXiv2013

A numerical methodology for enforcing maximum principles and the non-negative constraint for transient diffusion equations

Transient diffusion equations arise in many branches of engineering and applied sciences (e.g., heat transfer and mass transfer), and are parabolic partial differential equations. It is well-known that, under certain assumptions on the input data, these equations satisfy important mathematical properties like maximum principles and the non-negative constraint, which have implications in mathematical modeling. However, existing numerical formulations for these types of equations do not, in general, satisfy maximum principles and the non-negative constraint. In this paper, we present a methodology for enforcing maximum principles and the non-negative constraint for transient anisotropic diffusion equation. The method of horizontal lines (also known as the Rothe method) is applied in which the time is discretized first. This results in solving steady anisotropic diffusion equation with decay equation at every discrete time level. The proposed methodology for transient anisotropic diffusion equation will satisfy maximum principles and the non-negative constraint on general computational grids, and with no additional restrictions on the time step. We illustrate the performance and accuracy of the proposed formulation using representative numerical examples. We also perform numerical convergence of the proposed methodology. For comparison, we also present the results from the standard single-field semi-discrete formulation and the results from a popular software package, which all will violate maximum principles and the non-negative constraint.

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arXiv2013

A numerical framework for diffusion-controlled bimolecular-reactive systems to enforce maximum principles and non-negative constraint

We present a novel computational framework for diffusive-reactive systems that satisfies the non-negative constraint and maximum principles on general computational grids. The governing equations for the concentration of reactants and product are written in terms of tensorial diffusion-reaction equations. % We restrict our studies to fast irreversible bimolecular reactions. If one assumes that the reaction is diffusion-limited and all chemical species have the same diffusion coefficient, one can employ a linear transformation to rewrite the governing equations in terms of invariants, which are unaffected by the reaction. This results in two uncoupled tensorial diffusion equations in terms of these invariants, which are solved using a novel non-negative solver for tensorial diffusion-type equations. The concentrations of the reactants and the product are then calculated from invariants using algebraic manipulations. The novel aspect of the proposed computational framework is that it will always produce physically meaningful non-negative values for the concentrations of all chemical species. Several representative numerical examples are presented to illustrate the robustness, convergence, and the numerical performance of the proposed computational framework. We will also compare the proposed framework with other popular formulations. In particular, we will show that the Galerkin formulation (which is the standard single-field formulation) does not produce reliable solutions, and the reason can be attributed to the fact that the single-field formulation does not guarantee non-negative solutions. We will also show that the clipping procedure (which produces non-negative solutions but is considered as a variational crime) does not give accurate results when compared with the proposed computational framework.

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arXiv2013

Modification to Darcy model for high pressure and high velocity applications and associated mixed finite element formulations

The Darcy model is based on a plethora of assumptions. One of the most important assumptions is that the Darcy model assumes the drag coefficient to be constant. However, there is irrefutable experimental evidence that viscosities of organic liquids and carbon-dioxide depend on the pressure. Experiments have also shown that the drag varies nonlinearly with respect to the velocity at high flow rates. In important technological applications like enhanced oil recovery and geological carbon-dioxide sequestration, one encounters both high pressures and high flow rates. It should be emphasized that flow characteristics and pressure variation under varying drag are both quantitatively and qualitatively different from that of constant drag. Motivated by experimental evidence, we consider the drag coefficient to depend on both the pressure and velocity. We consider two major modifications to the Darcy model based on the Barus formula and Forchheimer approximation. The proposed modifications to the Darcy model result in nonlinear partial differential equations, which are not amenable to analytical solutions. To this end, we present mixed finite element formulations based on least-squares formalism and variational multiscale formalism for the resulting governing equations. The proposed modifications to the Darcy model and its associated finite element formulations are used to solve realistic problems with relevance to enhanced oil recovery. We also study the competition between the nonlinear dependence of drag on the velocity and the dependence of viscosity on the pressure. To the best of the authors' knowledge such a systematic study has not been performed.

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arXiv2013

A mixed formulation for a modification to Darcy equation based on Picard linearization and numerical solutions to large-scale realistic problems

In this paper we consider a modification to Darcy equation by taking into account the dependence of viscosity on the pressure. We present a stabilized mixed formulation for the resulting governing equations. Equal-order interpolation for the velocity and pressure is considered, and shown to be stable (which is not the case under the classical mixed formulation). The proposed mixed formulation is tested using a wide variety of numerical examples. The proposed formulation is also implemented in a parallel setting, and the performance of the formulation for large-scale problems is illustrated using a representative problem. Two practical and technologically important problems, one each on enhanced oil recovery and geological carbon-dioxide sequestration, are solved using the proposed formulation. The numerical examples show that the predictions based on Darcy model are qualitatively and quantitatively different from that of the predictions based on the modified Darcy model, which takes into account the dependence of the viscosity on the pressure. In particular, the numerical example on the geological carbon-dioxide sequestration shows that Darcy model over-predicts the leakage into an abandoned well when compared to that of the modified Darcy model. On the other hand, the modified Darcy model predicts higher pressures and higher pressure gradients near the injection well. These predictions have dire consequences in predicting damage and fracture zones, and designing the seal, whose integrity is crucial to the safety of a geological carbon-dioxide sequestration geosystem.

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arXiv2013

A framework for coupling flow and deformation of the porous solid

In this paper, we consider the flow of an incompressible fluid in a deformable porous solid. We present a mathematical model using the framework offered by the theory of interacting continua. In its most general form, this framework provides a mechanism for capturing multiphase flow, deformation, chemical reactions and thermal processes, as well as interactions between the various physics in a conveniently implemented fashion. To simplify the presentation of the framework, results are presented for a particular model than can be seen as an extension of Darcy's equation (which assumes that the porous solid is rigid) that takes into account elastic deformation of the porous solid. The model also considers the effect of deformation on porosity. We show that using this model one can recover identical results as in the framework proposed by Biot and Terzaghi. Some salient features of the framework are as follows: (a) It is a consistent mixture theory model, and adheres to the laws and principles of continuum thermodynamics, (b) the model is capable of simulating various important phenomena like consolidation and surface subsidence, and (c) the model is amenable to several extensions. We also present numerical coupling algorithms to obtain coupled flow-deformation response. Several representative numerical examples are presented to illustrate the capability of the mathematical model and the performance of the computational framework.

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arXiv2011

On the performance of high-order finite elements with respect to maximum principles and the non-negative constraint for diffusion-type equations

The main aim of this paper is to document the performance of $p$-refinement with respect to maximum principles and the non-negative constraint. The model problem is (steady-state) anisotropic diffusion with decay (which is a second-order elliptic partial differential equation). We considered the standard single-field formulation (which is based on the Galerkin formalism) and two least-squares-based mixed formulations. We have employed non-uniform Lagrange polynomials for altering the polynomial order in each element, and we have used $p = 1, ..., 10$. It will be shown that the violation of the non-negative constraint will not vanish with $p$-refinement for anisotropic diffusion. We shall illustrate the performance of $p$-refinement using several representative problems. The intended outcome of the paper is twofold. Firstly, this study will caution the users of high-order approximations about its performance with respect to maximum principles and the non-negative constraint. Secondly, this study will help researchers to develop new methodologies for enforcing maximum principles and the non-negative constraint under high-order approximations.

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arXiv2011

A framework for coupled deformation-diffusion analysis with application to degradation/healing

This paper deals with the formulation and numerical implementation of a fully coupled continuum model for deformation-diffusion in linearized elastic solids. The mathematical model takes into account the effect of the deformation on the diffusion process, and the affect of the transport of an inert chemical species on the deformation of the solid. We then present a robust computational framework for solving the proposed mathematical model, which consists of coupled non-linear partial differential equations. It should be noted that many popular numerical formulations may produce unphysical negative values for the concentration, particularly, when the diffusion process is anisotropic. The violation of the non-negative constraint by these numerical formulations is not mere numerical noise. In the proposed computational framework we employ a novel numerical formulation that will ensure that the concentration of the diffusant be always non-negative, which is one of the main contributions of this paper. Representative numerical examples are presented to show the robustness, convergence, and performance of the proposed computational framework. Another contribution of this paper is to systematically study the affect of transport of the diffusant on the deformation of the solid and vice-versa, and their implication in modeling degradation/healing of materials. We show that the coupled response is both qualitatively and quantitatively different from the uncoupled response.

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arXiv2010

Variational structure of the optimal artificial diffusion method for the advection-diffusion equation

In this research note we provide a variational basis for the optimal artificial diffusion method, which has been a cornerstone in developing many stabilized methods. The optimal artificial diffusion method produces exact nodal solutions when applied to one-dimensional problems with constant coefficients and forcing function. We first present a variational principle for a multi-dimensional advective-diffusive system, and then derive a new stable weak formulation. When applied to one-dimensional problems with constant coefficients and forcing function, this resulting weak formulation will be equivalent to the optimal artificial diffusion method. We present representative numerical results to corroborate our theoretical findings.

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arXiv2010

On the performance of the variational multiscale formulation for subsurface flow and transport in heterogeneous porous media

The following work compares two popular mixed finite elements used to model subsurface flow and transport in heterogeneous porous media; the lowest order Raviart-Thomas element and the variational multiscale stabilized element. Comparison is made based on performance for several problems of engineering relevance that involve highly heterogenous material properties (permeability ratios of up to $1\times10^5$), open flow boundary conditions (pressure driven flows), and large scale domains in two dimensions. Numerical experiments are performed to show the degree to which mass conservation is violated when a flow field computed using either element is used as the advection velocity in a transport model. The results reveal that the variational multiscale element shows considerable mass production or loss for problems that involve flow tangential to layers of differing permeability, but marginal violation of local mass balance for problems of less orthogonality in the permeability. The results are useful in establishing rudimentary estimates of the error produced by using the variational mutliscale element for several different types of problems.

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arXiv2010

A stabilized mixed formulation for unsteady Brinkman equation based on the method of horizontal lines

In this paper, we present a stabilized mixed formulation for unsteady Brinkman equation. The formulation is systematically derived based on the variational multiscale formalism and the method of horizontal lines. The derivation does not need the assumption that the fine-scale variables do not depend on the time, which is the case with the conventional derivation of multiscale stabilized formulations for transient mixed problems. An expression for the stabilization parameter is obtained in terms of a bubble function, and appropriate bubble functions for various finite elements are also presented. Under the proposed formulation, equal-order interpolation for the velocity and pressure (which is computationally the most convenient) is stable. Representative numerical results are presented to illustrate the performance of the proposed formulation. Spatial and temporal convergence studies are also performed, and the proposed formulation performed well.

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arXiv2010

On the vibrations of lumped parameter systems governed by differential-algebraic equations

In this paper, we consider the vibratory motions of lumped parameter systems wherein the components of the system cannot be described by constitutive expressions for the force in terms of appropriate kinematical quantities. Such physical systems reduce to a system of differential-algebraic equations, which invariably need to be solved numerically. To illustrate the issues with clarity, we consider a simple system in which the dashpot is assumed to contain a "Bingham" fluid for which one cannot describe the force in the dashpot as a function of the velocity. On the other hand, one can express the velocity as a function of the force.

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arXiv2010

Enforcing the non-negativity constraint and maximum principles for diffusion with decay on general computational grids

In this paper, we consider anisotropic diffusion with decay, and the diffusivity coefficient to be a second-order symmetric and positive definite tensor. It is well-known that this particular equation is a second-order elliptic equation, and satisfies a maximum principle under certain regularity assumptions. However, the finite element implementation of the classical Galerkin formulation for both anisotropic and isotropic diffusion with decay does not respect the maximum principle. We first show that the numerical accuracy of the classical Galerkin formulation deteriorates dramatically with increase in the decay coefficient for isotropic medium and violates the discrete maximum principle. However, in the case of isotropic medium, the extent of violation decreases with mesh refinement. We then show that, in the case of anisotropic medium, the classical Galerkin formulation for anisotropic diffusion with decay violates the discrete maximum principle even at lower values of decay coefficient and does not vanish with mesh refinement. We then present a methodology for enforcing maximum principles under the classical Galerkin formulation for anisotropic diffusion with decay on general computational grids using optimization techniques. Representative numerical results (which take into account anisotropy and heterogeneity) are presented to illustrate the performance of the proposed formulation.

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arXiv2010

A numerical study of fluids with pressure dependent viscosity flowing through a rigid porous medium

In this paper we consider modifications to Darcy's equation wherein the drag coefficient is a function of pressure, which is a realistic model for technological applications like enhanced oil recovery and geological carbon sequestration. We first outline the approximations behind Darcy's equation and the modifications that we propose to Darcy's equation, and derive the governing equations through a systematic approach using mixture theory. We then propose a stabilized mixed finite element formulation for the modified Darcy's equation. To solve the resulting nonlinear equations we present a solution procedure based on the consistent Newton-Raphson method. We solve representative test problems to illustrate the performance of the proposed stabilized formulation. One of the objectives of this paper is also to show that the dependence of viscosity on the pressure can have a significant effect both on the qualitative and quantitative nature of the solution.

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arXiv2009

On dual Schur domain decomposition method for linear first-order transient problems

This paper addresses some numerical and theoretical aspects of dual Schur domain decomposition methods for linear first-order transient partial differential equations. In this work, we consider the trapezoidal family of schemes for integrating the ordinary differential equations (ODEs) for each subdomain and present four different coupling methods, corresponding to different algebraic constraints, for enforcing kinematic continuity on the interface between the subdomains. Method 1 (d-continuity) is based on the conventional approach using continuity of the primary variable and we show that this method is unstable for a lot of commonly used time integrators including the mid-point rule. To alleviate this difficulty, we propose a new Method 2 (Modified d-continuity) and prove its stability for coupling all time integrators in the trapezoidal family (except the forward Euler). Method 3 (v-continuity) is based on enforcing the continuity of the time derivative of the primary variable. However, this constraint introduces a drift in the primary variable on the interface. We present Method 4 (Baumgarte stabilized) which uses Baumgarte stabilization to limit this drift and we derive bounds for the stabilization parameter to ensure stability. Our stability analysis is based on the ``energy'' method, and one of the main contributions of this paper is the extension of the energy method (which was previously introduced in the context of numerical methods for ODEs) to assess the stability of numerical formulations for index-2 differential-algebraic equations (DAEs).

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arXiv2009

Non-negative mixed finite element formulations for a tensorial diffusion equation

We consider the tensorial diffusion equation, and address the discrete maximum-minimum principle of mixed finite element formulations. In particular, we address non-negative solutions (which is a special case of the maximum-minimum principle) of mixed finite element formulations. The discrete maximum-minimum principle is the discrete version of the maximum-minimum principle. In this paper we present two non-negative mixed finite element formulations for tensorial diffusion equations based on constrained optimization techniques (in particular, quadratic programming). These proposed mixed formulations produce non-negative numerical solutions on arbitrary meshes for low-order (i.e., linear, bilinear and trilinear) finite elements. The first formulation is based on the Raviart-Thomas spaces, and is obtained by adding a non-negative constraint to the variational statement of the Raviart-Thomas formulation. The second non-negative formulation based on the variational multiscale formulation. For the former formulation we comment on the affect of adding the non-negative constraint on the local mass balance property of the Raviart-Thomas formulation. We also study the performance of the active set strategy for solving the resulting constrained optimization problems. The overall performance of the proposed formulation is illustrated on three canonical test problems.

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arXiv2008

A variational multiscale Newton-Schur approach for the incompressible Navier-Stokes equations

In the following paper, we present a consistent Newton-Schur solution approach for variational multiscale formulations of the time-dependent Navier-Stokes equations in three dimensions. The main contributions of this work are a systematic study of the variational multiscale method for three-dimensional problems, and an implementation of a consistent formulation suitable for large problems with high nonlinearity, unstructured meshes, and non-symmetric matrices. In addition to the quadratic convergence characteristics of a Newton-Raphson based scheme, the Newton-Schur approach increases computational efficiency and parallel scalability by implementing the tangent stiffness matrix in Schur complement form. As a result, more computations are performed at the element level. Using a variational multiscale framework, we construct a two-level approach to stabilizing the incompressible Navier-Stokes equations based on a coarse and fine-scale subproblem. We then derive the Schur complement form of the consistent tangent matrix. We demonstrate the performance of the method for a number of three-dimensional problems for Reynolds number up to 1000 including steady and time-dependent flows.

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arXiv2008

A stabilized finite element formulation for advection-diffusion using the generalized finite element framework

The following work presents a generalized (extended) finite element formulation for the advection-diffusion equation. Using enrichment functions that represent the exponential nature of the exact solution, smooth numerical solutions are obtained for problems with steep gradients and high Peclet numbers (up to Pe = 25) in one and two-dimensions. As opposed to traditional stabilized methods that require the construction of stability parameters and stabilization terms, the present work avoids numerical instabilities by improving the classical Galerkin solution with an enrichment function. To contextualize this method among other stabilized methods, we show by decomposition of the solution (in a multiscale manner) an equivalence to both Galerkin/least-squares type methods and those that use bubble functions. This work also presents a strategy for constructing the enrichment function for problems with complex geometries by employing a global-local approach.

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arXiv2008

Consistent Newton-Raphson vs. fixed-point for variational multiscale formulations for incompressible Navier-Stokes

The following paper compares a consistent Newton-Raphson and fixed-point iteration based solution strategy for a variational multiscale finite element formulation for incompressible Navier-Stokes. The main contributions of this work include a consistent linearization of the Navier-Stokes equations, which provides an avenue for advanced algorithms that require origins in a consistent method. We also present a comparison between formulations that differ only in their linearization, but maintain all other equivalences. Using the variational multiscale concept, we construct a stabilized formulation (that may be considered an extension of the MINI element to nonlinear Navier-Stokes). We then linearize the problem using fixed-point iteration and by deriving a consistent tangent matrix for the update equation to obtain the solution via Newton-Raphson iterations. We show that the consistent formulation converges in fewer iterations, as expected, for several test problems. We also show that the consistent formulation converges for problems for which fixed-point iteration diverges. We present the results of both methods for problems of Reynold's number up to 5000.

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