h,p,k MATHEMATICAL AND COMPUTATIONAL FINITE ELEMENT FRAMEWORK WITH VARIATIONALLY CONSISTENT INTEGRAL FORMS FOR BOUNDARY VALUE AND INITIAL VALUE PROBLEMS
This paper presents a unified mathematical and computational finite element framework for boundary value problems (BVPs) and initial value problems (IVPs) in which (i) the integral forms are ensured to yield algebraic systems that guarantee unconditionally stable computations (ii) the local approximations are considered in h, p, k higher order approximation spaces in which h is the characteristic length, p is the degree of local approximation and k is the order of the approximation space. The order k of the approximation space ensures global differentiability of the approximations of order (k - 1). Surana et al have shown that k, the order of the approximation space, is an independent computational parameter in all finite element processes in addition to h and p and hence the name h.p.k framework instead of h, p framework used currently, and k-version of the finite element method in addition to h- and p-versions used presently. Introduction of k and hence global differentiability of approximations of order (k - 1) permits the use of partial differential equations containing higher order derivatives of the dependent variables in the integral forms while maintaining the integrals over the discretizations in the Riemann sense. The benefits being the desired physics in the computational processes as well as the ability to incorporate higher order global differentiability features of the theoretical solution in the design of the computational process. In order to address finite element computations of all BVPs and IVPs in a rigorous and application independent fashion and without resorting to ad-hoc problem dependent special treatments, we consider the totality of all BVPs and IVPs and classify the differential operators appearing in them mathematically. This mathematical classification is then used in conjunction with methods of approximation to determine which methods of approximation for which classes of differential operators yield unconditionally stable computational processes. We consider specific details for BVPs and IVPs in the following. Surana et al. have shown that differential operators appearing in all BVPs can be mathematically classified in three categories: self-adjoint, non-self adjoint and non-linear. When methods of approximation such as Galerkin method (GM), Petrov-Galerkin method (PGM), weighted residual method (WRM). Galerkin method with weak form (GM/WF) and least squares process (LSP) based on error functional are used for these three classes of operators we obtain integral forms (necessary conditions) that yield algebraic systems upon substituting local approximations in them. Authors in Ref.l-3 established a link or correspondence between the integral forms resulting from the methods of approximation and the elements of the calculus of variations to determine which integral forms yield unconditionally stable computations. This leads to the definitions of variationally consistent (VC) and variationally inconsistent (VIC) integral forms. VC integral forms are in conformity with all aspects of the calculus of variations whereas VIC integral forms violate at least one or more aspects or conditions resulting from it. VC integral forms yield symmetric algebraic systems with positive definite coefficient matrices that have positive eigenvalues and real basis and hence resulting in unconditionally stable computations.On the other hand. VIC forms yield non-symmetric coefficient matrices that may have non-positive or complex eigenvalues and partially or completely complex basis and are not always ensured to be positive definite. This approach establishes which integral forms are worthy of consideration for which classes of differential operators in the development of unconditionally stable finite element processes for BVPs. Surana et al. have also shown that in case of IVPs, only space-time coupled finite element processes are worthy of consideration if we are seeking a general and mathematically rigorous finite element computational framework for all IVPs. Authors in have shown that space-time differential operators can be mathematically classified over the entire space-time domain of the IVP or over a space-time strip or slab for an increment of time in two categories: non-self adjoint and non-linear. Applications of space-time approximation methods such as space-time GM.PGM, WRM,GM/WF,LSP yield space-time integral forms (necessary conditions) that result in algebraic systems upon substituting space-time local approximations. By establishing a correspondence between the space-time integral forms resulting from the space-time methods of approximation and the elements of the calculus of variations, we can establish which integral forms yield unconditionally stable computations. By introducing definitions of space-time variationally consistent (STVC) integral forms (in which integral forms conform to the elements of the calculus of variations) and space-time variationally inconsistent (STVIC) integral forms (in which integral forms are in violation of one or more conditions resulting from the calculus of variations), the authors have shown that STVC integral forms yield unconditionally stable computations during the entire evolution. STVIC integral forms do not always ensure unconditionally stable computations. Similar to VC and VIC integral forms for BVPs, in this case also. STVC space-time integral forms yield symmetric positive definite coefficient matrices that have positive real eigenvalues and real basis. Whereas STVIC space-time integral forms yield non-symmetric coefficient matrices that may have negative or complex eigenvalues and the basis may be partially or completely complex. Thus, by using this approach, it is possible to design space-time finite element processes for the two classes of space-time differential operators that are ensured to yield unconditionally stable computations. Use of h,p,k framework with higher order approximation spaces permits desired global differentiability of approximations in space as well as time. Minimally conforming approximation spaces, criteria for their selection and their importance in designing finite element computations for BVPs and IVPs are discussed and illustrated. VC and STVC integral forms with local approximation in higher order h. p, k spaces provide the most general and problem independent framework for finite element processes for all BVPs and IVPs regardless of their origin, application or complexity of the mathematical models. This is a straight forward but mathematically consistent and rigorous methodology that is completely free of ad-hoc problem dependent treatments such as upwinding methods. In the following we present details for BVPs and IVPs followed by concluding remarks.
h, p,k framework k-version variantional consistency Galerkin method with weak form least squares
K. S. Surana J. N. Reddy A. Romkes
Mechanical Engineering, University ofKansas,Lawrence.Kansas, USA Mechanical Engineering, Texas A&M University, College Station . Texas, USA Mechanical Engineering, University of Kansas,Lawrence.Kansas, USA
国际会议
广州
英文
12-25
2010-12-17（万方平台首次上网日期，不代表论文的发表时间）