k-Version of Finite Element Method for BVPs and IVPs
| Autor: | Sri Sai Charan Mathi, Karan S. Surana, Celso H. Carranza |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
General Mathematics
finite element method MathematicsofComputing_GENERAL higher order global differentiability 02 engineering and technology Isogeometric analysis 01 natural sciences k-version 0203 mechanical engineering Convergence (routing) QA1-939 Computer Science (miscellaneous) Applied mathematics Initial value problem isogeometric Boundary value problem 0101 mathematics Engineering (miscellaneous) Mathematics tensor product higher order spaces Differential operator IVPs Finite element method variational consistency 010101 applied mathematics BVPs TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES 020303 mechanical engineering & transports Tensor product Self-adjoint operator |
| Zdroj: | Mathematics Volume 9 Issue 12 Mathematics, Vol 9, Iss 1333, p 1333 (2021) |
| ISSN: | 2227-7390 |
| DOI: | 10.3390/math9121333 |
| Popis: | The paper presents k-version of the finite element method for boundary value problems (BVPs) and initial value problems (IVPs) in which global differentiability of approximations is always the result of the union of local approximations. The higher order global differentiability approximations (HGDA/DG) are always p-version hierarchical that permit use of any desired p-level without effecting global differentiability. HGDA/DG are true Ci, Cij, Cijk, hence the dofs at the nonhierarchical nodes of the elements are transformable between natural and physical coordinate spaces using calculus. This is not the case with tensor product higher order continuity elements discussed in this paper, thus confirming that the tensor product approximations are not true Ci, Cijk, Cijk approximations. It is shown that isogeometric analysis for a domain with more than one patch can only yield solutions of class C0. This method has no concept of finite elements and local approximations, just patches. It is shown that compariso of this method with k-version of the finite element method is meaningless. Model problem studies in R2 establish accuracy and superior convergence characteristics of true Cijp-version hierarchical local approximations presented in this paper over tensor product approximations. Convergence characteristics of p-convergence, k-convergence and pk-convergence are illustrated for self adjoint, non-self adjoint and non-linear differential operators in BVPs. It is demonstrated that h, p and k are three independent parameters in all finite element computations. Tensor product local approximations and other published works on k-version and their limitations are discussed in the paper and are compared with present work. |
| Databáze: | OpenAIRE |
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