Variational Consistency Research Articles

A quasi-conforming embedded reproducing kernel particle method for heterogeneous materials

We present a quasi-conforming embedded reproducing kernel particle method (QCE-RKPM) for modeling heterogeneous materials that makes use of techniques not available to mesh-based methods such as the finite element method (FEM) and avoids many of the drawbacks in current embedded and immersed formulations which are based on meshed methods. The different material domains are discretized independently thus avoiding time-consuming, conformal meshing. In this approach, the superposition of foreground (inclusion) and background (matrix) domain integration smoothing cells are corrected by a quasi-conforming quadtree subdivision on the background integration smoothing cells. Due to the non-conforming nature of the background integration smoothing cells near the material interfaces, a variationally consistent (VC) correction for domain integration is introduced to restore integration constraints and thus optimal convergence rates at a minor computational cost. Additional interface integration smoothing cells with area (volume) correction, while non-conforming, can be easily introduced to further enhance the accuracy and stability of the Galerkin solution using VC integration on non-conforming cells. To properly approximate the weak discontinuity across the material interface by a penalty-free Nitsche’s method with enhanced coercivity, the interface nodes on the surface of the foreground discretization are also shared with the background discretization. As such, there are no tunable parameters, such as those involved in the penalty type method, to enforce interface compatibility in this approach. The advantage of this meshfree formulation is that it avoids many of the instabilities in mesh-based immersed and embedded methods. The effectiveness of QCE-RKPM is illustrated with several examples.

Stabilized and variationally consistent integrated meshfree formulation for advection-dominated problems

Meshfree methods such as reproducing kernel (RK) approximation are suitable for modeling fluid flow problems because of their flexibility in controlling local smoothness and order of basis, as well as straightforward construction of higher-order gradients. However, Eulerian-described partial differential equations often suffer from numerical instability in the solution of Bubnov–Galerkin​ methods because of the non-self-adjoint advection terms. This is true for both mesh-based and meshfree methods. Although stabilized Petrov–Galerkin formulation, such as the variational multiscale method, has addressed this issue, a careless selection of the numerical quadrature can still result in variational inconsistency in the Galerkin weak form, which leads to a suboptimal convergence. Strong advection could also amplify the unstable modes from a reduced quadrature. This study provides a variationally consistent (VC) approach to correct the loss of Galerkin exactness in nodally integrated meshfree modeling for the advection diffusion equation. A gradient stabilization method is proposed to enhance the coercivity of the system. Several numerical examples are provided to verify the effectiveness and efficiency of the proposed approaches in modeling advection dominated problems.

Finite Element Processes Based on GM/WF in Non-Classical Solid Mechanics

In non-classical thermoelastic solids incorporating internal rotation and conjugate Cauchy moment tensor the mechanical deformation is reversible. This suggests that within the realm of linear mathematical models that only consider small strains and small deformation the mechanical deformation is reversible. Hence, it is possible to recast the conservation and balance laws along with constitutive theories in a form that adjoint A* of the differential operator A in mathematical model is same as the differential operator A. This holds regardless of whether we consider an initial value problem (IVP) (when the integrals over open boundary are neglected) or boundary value problem (BVP). Thus, in such cases Galerkin method with weak form (GM/WF) for BVPs and space-time Galerkin method with weak form (STGM/WF) for IVPs are highly meritorious due to the fact that: 1) the integral form for BVPs is variationally consistent (VC) and 2) the space-time integral forms for IVP are space time variationally consistent (STVC). The consequence of VC and STVC integral forms is that the resulting coefficient matrices are symmetric and positive definite ensuring unconditionally stable computational processes for both BVPs and IVPs. Other benefits of GM/WF and space-time GM/WF are simplicity of specifying boundary conditions and initial conditions, especially traction boundary conditions and initial conditions on curved boundaries due to self-equilibrating nature of the sum of secondary variables that only exist in GM/WF due to concomitant. In fact, zero traction conditions are automatically satisfied in GM/WF, hence need not be specified at all. While VC and STVC feature also exists in least squares process (LSP) and space-time least squares finite element processes (STLSP) for BVPs and IVPs, the ease of specifying traction boundary conditions feature in GM/WF and STGM/WF is highly meritorious compared to LSP and STLSP in which zero traction conditions need to be explicitly specified. A disadvantage of GM/WF and STGM/ WF is that the mathematical models (momentum equations) needed in the desired form contain higher order derivatives of displacements (upto fourth order), hence necessitate use of higher order spaces in their solution. As well known, this problem can be easily overcome in LSP and STLSP by introduction of auxiliary equations and auxiliary variables, thus keeping the highest orders of the derivatives of the dependent variables to one or any other desired order. A serious disadvantage of this approach in LSP is the significant increase in the number of dependent variables, hence poor computational efficiency. In this paper we consider non-classical continuum models for internally polar linear elastic solids in which internal rotations due to displacement gradient tensor (hence internal polar physics) are considered in the conservation and the balance laws and the constitutive theories. For simplicity, we only consider isothermal case; hence energy equation is not part of mathematical model. When using mathematical models derived in displacements in GM/WF and LSP in constructing integral forms, we note that in GM/WF the number of dependent variables is reduced drastically (only three in R3), whereas in case of first order systems used in LSP and STLSP we may have as many as 22 dependent variables for isothermal case. Thus, GM/WF results in dramatic improvement in computational efficiency as well as accuracy when minimally conforming spaces are used for approximations. In this paper we only consider mathematical model in R2 for BVPs (for simplicity). Mathematical models for IVP and BVP in R3 will be considered in subsequent paper. The integral form is derived in R2 using GM/WF. Numerical examples are presented using GM/WF and LSP to demonstrate advantages of finite element process derived using integral form based on GM/WF for non-classical linear theories for solids incorporating internal rotations due to displacement gradient tensor.

Error Estimations, Error Computations, and Convergence Rates in FEM for BVPs

This paper presents derivation of a priori error estimates and convergence rates of finite element processes for boundary value problems (BVPs) described by self adjoint, non-self adjoint, and nonlinear differential operators. A posteriori error estimates are discussed in context with local approximations in higher order scalar product spaces. A posteriori error computational framework (without the knowledge of theoretical solution) is presented for all BVPs regardless of the method of approximation employed in constructing the integral form. This enables computations of local errors as well as the global errors in the computed finite element solutions. The two most significant and essential aspects of the research presented in this paper that enable all of the features described above are: 1) ensuring variational consistency of the integral form(s) resulting from the methods of approximation for self adjoint, non-self adjoint, and nonlinear differential operators and 2) choosing local approximations for the elements of a discretization in a subspace of a higher order scalar product space that is minimally conforming, hence ensuring desired global differentiability of the approximations over the discretizations. It is shown that when the theoretical solution of a BVP is analytic, the a priori error estimate (in the asymptotic range, discussed in a later section of the paper) is independent of the method of approximation or the nature of the differential operator provided the resulting integral form is variationally consistent. Thus, the finite element processes utilizing integral forms based on different methods of approximation but resulting in VC integral forms result in the same a priori error estimate and convergence rate. It is shown that a variationally consistent (VC) integral form has best approximation property in some norm, conversely an integral form with best approximation property in some norm is variationally consistent. That is best approximation property of the integral form and the VC of the integral form is equivalent, one cannot exist without the other, hence can be used interchangeably. Dimensional model problems consisting of diffusion equation, convection-diffusion equation, and Burgers equation described by self adjoint, non-self adjoint, and nonlinear differential operators are considered to present extensive numerical studies using Galerkin method with weak form (GM/WF) and least squares process (LSP) to determine computed convergence rates of various error norms and present comparisons with the theoretical convergence rates.

Computations of Numerical Solutions in Polymer Flows Using Giesekus Constitutive Model in the hpk Framework with Variationally Consistent Integral Forms

This paper presents numerical solutions of boundary value problems (BVPs) for 1-D and 2-D polymer flows using the Giesekus constitutive model in the hpk mathematical and finite element (FE) computational framework utilizing variationally consistent (VC) integral forms. In the mathematical framework used here, h , the characteristic length, p , the degree of local approximation, and k , the order of the approximation space, are independent parameters as opposed to h and p used currently. The order k of the approximation space allows local approximations of higher order global differentiability. The VC integral forms ensure unconditionally stable computational processes, hence eliminating the need for the currently used upwinding methods for stabilizing computations. The work presented in this paper focuses on the following major-areas of investigation using the hpk framework and VC integral forms: (1) investigations of the different choices of stresses as dependent variables in the mathematical models on the performance of the resulting computational processes; (2) computations of numerical solutions for higher Deborah numbers; (3) solutions that are independent of the hpk computational parameters for fixed physics. 1-D fully developed flow, 2-D developing flow, and lid-driven cavity are used as model problems in the numerical studies.

Numerical Simulations of BVPs and IVPs in Fiber Spinning Using Giesekus Constitutive Model in hpk Framework

This paper presents numerical simulations of 1-D boundary value problems (BVPs) and initial value problems (IVPs) in fiber spinning using Giesekus constitutive model in hpk mathematical and computational framework with variationally consistent (VC) and space-time variationally consistent (STVC) integral forms. In hpk framework, h, the characteristic length, p, the degree of local approximations, k, the order of the approximation space in space as well as space and time, are independent variables in all finite element (FE) computational processes as opposed to h and p in currently used FE processes. k, the order of the approximation space, allows higher order global differentiability local approximations in space as well as in time that are necessitated by the mathematical models and the higher order global differentiability features of the theoretical solutions. VC integral forms for the BVPs and STVC integral forms for IVPs yield unconditionally stable computational processes. It is shown that for 1-D BVPs and IVPs the mathematical models used in the published work for fiber spinning are deficient in incorporating the desired physics, utilize redundant dependent variables, and hence can lead to spurious numerical solutions. Within the assumptions used in the literature, a new mathematical model is presented in this paper that is free of the problems described here. Numerical studies are presented for BVPs and IVPs for four different fluids described by the Giesekus constitutive model for high draw ratios. In all numerical studies presented in the paper, the stationary states of the evolutions are shown to be in extremely good agreement with the solutions of the corresponding BVPs.

Numerical Solutions of BVPs in 2-D Viscous Compressible Flows Using hpk Framework

This paper presents numerical simulations of the two dimensional boundary value problems (BVPs) in viscous compressible flows using finite element method in hpk framework [1]–[2] in which the integral forms are variationally consistent (VC). The mathematical models utilize Navier-Stokes equations incorporating physical viscosity, conductivity and other transport properties. The finite element method based on hpk framework permits higher order global differentiability approximation and the variationally consistent integral forms [1]–[2] ensure unconditionally stable computations for all choices hpk and the dimensionless parameters in the mathematical models and thus do not require the use of upwinding methods such as SUPG/DC/LS [3]–[4]. Mach 1, 2, 3 and 5 flows over a flat plate (Carter's plate) is used as a model problems with ideal gas law.

K-Version of finite element method in 2D-polymer flows: Upper convected Maxwell model

In this paper, a new mathematical framework based on h, p, k and variational consistency (VC) of the integral forms is utilized to develop a finite element computational process for steady two dimensional polymer flows with upper convected Maxwell constitutive model (UCMM). Alternate forms of choices of dependent variables in the governing differential equations (GDEs) are considered and it is concluded that u, v, p, τ choice yielding strong form of the GDEs is meritorious. Since, the differential operators for all possible choices of dependent variables are non-linear, Galerkin method and Galerkin method with weak form are variationally inconsistent (VIC). The coefficient matrices in these processes are non-symmetric and hence may have partial or completely complex basis and the resulting computational processes may yield spurious solutions. Furthermore, since the VC of VIC integral forms cannot be restored through any mathematically justifiable means, the computational processes in these approaches may remain spurious. Least squares processes utilizing GDEs in u, v, p, τ or any other variables are always variationally consistent. The coefficient matrices are symmetric, positive definite and hence always have real basis and thus naturally yield computational processes that are free of spurious solutions. The theoretical solution of GDEs are generally of higher order global differentiability. Numerical simulations of such solutions in which higher order global differentiability characteristics of the theoretical solutions are preserved, undoubtedly require local approximations in scalar product spaces H k , p ( Ω ¯ xy e ) containing higher order global differentiability local approximations. LSP with local approximation in H k , p ( Ω ¯ xy e ) spaces provides an excellent mathematical and computational framework in which it is possible to incorporate desired characteristics of the theoretical solution in the computational process. Numerical studies are presented for fully developed flow between parallel plates and a lid driven square cavity. M1 fluid is used in all numerical studies. The range of applicability of UCMM or lack of it is examined for both model problems for increasing De. A regularization of the velocity of the lid at the corners where stationary wall meets the lid is presented and is shown to simulate the real physics when the local approximations are in higher order spaces and when h d → 0. For both model problems shear rate γ ˙ ˆ is examined in the flow domain to establish validity of the UCMM constitutive model.

Galerkin/Least-Squares Finite Element Processes for BVP in h, p, k Mathematical Framework

This paper presents an investigation of the details of mathematical and computational aspects of the Galerkin/least-squares and the Galerkin/weak form least-squares finite element process within the mathematical and computational framework [1, 2, 3] based on h, p, k as independent computational parameters and requiring that the integral forms be variationally consistent (VC). Higher-order global differentiability of order (k−1) defined by the order k of the approximation space is essential for incorporating correct physics of the processes in the computations and that k is an independent parameter in addition to h and p in all finite element computations. In this paper the attributes of the Galerkin method, the Galerkin method with weak form, and least-squares processes in h, p, k framework with variationally consistent (VC) or variationally inconsistent (VIC) integral forms are utilized to investigate the mathematical features of the Galerkin/least-squares processes (GAL/LSP) and Galerkin/weak form least-squares process (GAL/WF/LSP) to establish (1) when such processes have a sound mathematical basis (2) role of functionals resulting from the Galerkin method, the Galerkin method with weak form and least-squares processes in GAL/LSP and GAL/WF/LSP (3) Importance of minimally conforming spaces and role of higher order spaces in GAL/LSP and GAL/WF/LSP. It is concluded that GAL/LSP only have sound mathematical basis for self adjoint differential operators when viewed within h,p,k framework with the requirement that the integral forms be variationally consistent. For non-self adjoint and non-linear differential operators GAL/LSP and GAL/WF/LSP do not have a sound mathematical basis within the proposed framework. Currently used finite element processes based on GAL/WF/LSP utilizing local approximations of class C 0 violate basic mathematical principles, are in violation with physics and hence do not provide a valid and mathematically sound approach. Investigation of the GAL/LSP and GAL/WF/LSP when the local approximations are in minimally conforming spaces or in spaces of order higher than minimally conforming spaces, shows that the functionals from GAL and GAL/WF processes only have detrimental affect on the LSP process. Numerical studies are presented to demonstrate various aspects and features.

Galerkin and Least-Squares Finite Element Processes for 2-D Helmholtz Equation in h, p, k Framework

Mathematical and computational details of the finite element processes using the Galerkin method, Galerkin method with weak form, and least-squares processes (LSP) for 2-D Helmholtz equation over a square domain with Dirichlet and Neumann boundary conditions are presented in h, p, k mathematical and computational framework. The order of the approximation space, k, defines the order of the global differentiability (k − 1) of the approximations, and it is an independent parameter, in addition to h and p, in the finite element computations. The discussion of variational consistency (VC) and variational inconsistency (VIC) of the integral forms resulting from the Galerkin method, the Galerkin method with weak form and least-squares processes and their impact on the resulting computational processes are presented. Higher-order global differentiability of the approximations is necessitated by the higher-order global differentiability of the theoretical solutions when they are analytic. Such solutions can be accurately simulated in h, p, k framework up to any desired order global differentiability. Numerical studies are presented for solutions of various classes using the Galerkin method with weak form and least square processes in h, p, k framework. Numerical studies demonstrate various features of the finite element processes and substantiate these observations.

K-Version Least Squares Finite Element Processes for 2-D Generalized Newtonian Fluid Flows

In this paper, a mathematical and computational framework based on h, p, k and variationally consistent (VC) integral form is utilized to present a finite element computational process for two dimensional steady, isothermal as well as non-isothermal fluid flows for power law and Carreau models of viscosity. Strong form of the governing differential equations is presented and utilized in dependent variables P, u, v and T. Since the differential operator is non-linear, Galerkin method, Petrov Galerkin method, method of weighted residuals and Galerkin method with weak forms yield integral form that are variationally inconsistent (VIC). Least Squares Processes on the other hand, yield integral forms that are VC. Computational processes based on VC integral forms are unconditionally stable and non-degenerate and hence do not require the use of problem dependent upwinding methods. The parameter k controlling global differentiability of the approximation permits higher order global approximations, which are necessitated by the physics in governing differential equations (GDEs). The behavior of the power law and Carreau models of shear thinning fluid behavior is investigated using theoretical solutions (when possible) as well as numerical simulations using the proposed methodology. Fully developed flow between parallel plates, steady pressure driven Couette flow and lid driven cavity are used as model problems.

H, p, k Least Squares Finite Element Processes for 1-D Helmholtz Equation

The paper presents mathematical details of the finite element processes using the Galerkin method with weak form and least square processes for 1-D Helmholtz equation with Robin boundary condition in h,p,k framework. The concepts of variational consistency (VC) and variational inconsistency (VIC) are discussed. It is shown that Galerkin method with or without weak form is variationally inconsistent (VIC). The VIC of the Galerkin method is due to non-zero wave number (κ term) as well as Robin boundary condition. These two aspects of VIC of the integral forms and their consequences are investigated individually as well as jointly. It is shown that the integral forms yield non-symmetric functional B(.,.) and hence the possibility of complex basis for the coefficient matrix for some choices of h, p and k, which may lead to spurious or oscillatory non-physical numerical solutions. It is demonstrated that the variational consistency of the Galerkin method or galerkin method with weak form cannot be restored through any mathematically justifiable means, hence spuriousness in the computed solution is inevitable for those ranges of physical and computational parameters for which basis of the coefficient matrix becomes complex, though may possibly be minimized using different approaches reported in the literature. Least square processes, on the other hand, are always variationally consistent for any wave number and any choice of computational parameters. Hence, the resulting coefficient matrices are always symmetric, positive definite and have real basis and thus are free of spurious numerical solutions. Periodic theoretical solutions of most model problems, including those used here, are of higher order global differentiability and hence necessitate use of higher order spaces for local approximations. Mathematical details as well as numerical studies are presented to illustrate various features of the proposed approach.

K‐Version of finite element method in 2‐D polymer flows: Oldroyd‐B constitutive model

AbstractIn this paper, a new mathematical framework based on h, p, k and variational consistency (VC) of the integral forms is utilized to develop a finite element computational process of two‐dimensional polymer flows utilizing Oldroyd‐B constitutive model. Alternate forms of the choices of dependent variables in the governing differential equations (GDEs) are considered and is concluded that u, v, p, τ choice yielding strong form of the GDEs is meritorious over others. It is shown that: (a) since, the differential operator in the GDEs is non‐linear, Galerkin method and Galerkin method with weak form are variationally inconsistent (VIC). The coefficient matrices in these processes are non‐symmetric and hence may have partial or completely complex basis and thus the resulting computational processes may be spurious. (b) Since the VC of the VIC integral forms cannot be restored through any mathematically justifiable means, the computational processes in these approaches always have possibility of spurious solutions. (c) Least squares process utilizing GDEs in u, v, p, τ (strong form of the GDEs) variables (as well as others) is variationally consistent. The coefficient matrices are always symmetric and positive definite and hence always have a real basis and thus naturally yield computational processes that are free of spurious solutions. (d) The theoretical solution of the GDEs are generally of higher order global differentiability. Numerical simulations of such solutions in which higher order global differentiability characteristics of the theoretical solution are preserved, undoubtedly requires local approximations in higher order scalar product spaces ${H}^{k,p}(\bar{\Omega}_{e})$. (e) LSP with local approximations in ${H}^{k,p}(\bar{\Omega}_{e})$ spaces provide an incomparable mathematical and computational framework in which it is possible to preserve desired characteristics of the theoretical solution in the computational process. Numerical studies are presented for fully developed flow between parallel plates and a lid driven square cavity. M1 fluid is used in all numerical studies. The range of applicability of the Oldroyd‐B model or lack of it is examined for both model problems for increasing De. A mathematical idealization of the corners where stationary wall meets the lid is presented and is shown to simulate the real physics when the local approximations are in higher order spaces and when hd→0. For both model problems shear rate $\hat{\dot{\gamma}}$ is examined in the flow domain to establish validity of the Oldroyd‐B constitutive model. Copyright © 2006 John Wiley & Sons, Ltd.

THE K-VERSION OF FINITE ELEMENT METHOD FOR NONLINEAR OPERATORS IN BVP

In the companion papers [1,2], authors introduced the concepts of k-version of finite element method and k, hk, pk, hkp-processes of the finite element method for boundary value problems described by self-adjoint and non-self adjoint operators using Ĥk,p(Ω) spaces with specific details including numerical studies for weak forms and least square processes. It was demonstrated that a variationally consistent (VC) weak form is possible when the differential operator is self-adjoint, however, in case of non-self-adjoint operators the weak forms are variationally inconsistent (VIC) which lead to degenerate computational processes that can produce spurious oscillations in the computed solutions. In this paper we demonstrate that when the boundary value problems are described by non-linear differential operators, Galerkin processes and weak forms can never be variationally consistent and hence result in degenerate computational processes and suffer from same problems as in the case of non-self-adjoint operators ...