Large elastic deformation of micromorphic shells. Part II. Isogeometric analysis

2019 ◽  
Vol 24 (12) ◽  
pp. 3753-3778 ◽  
Author(s):  
Amir Norouzzadeh ◽  
Reza Ansari ◽  
Mansour Darvizeh
Keyword(s):  
Finite Element ◽  
Elastic Deformation ◽  
Isogeometric Analysis ◽  
Constitutive Relations ◽  
Kinematic Model ◽  
Benchmark Problems ◽  
Element Analysis ◽  
Deformation Problem ◽  
Macro Scale ◽  
Large Elastic Deformation

In Part I of this study, a variational formulation was presented for the large elastic deformation problem of micromorphic shells. Using the novel matrix-vector format presented for the kinematic model, constitutive relations, and energy functions, an isogeometric analysis (IGA)-based solution strategy is developed, which appropriately estimates the macro- and micro-deformation field components. Due to the capability of constructing exact geometries and the powerful mesh refinement tools, IGA can be successfully applied to solve the equilibrium equations with dominant nonlinear terms. It is known that different types of locking phenomena take place in the conventional finite element analysis of thin shells based on low-order elements. Non-standard finite element models with mixed interpolation schemes and additional degrees of freedom (DOFs) or the ones used the high-order Lagrangian shell elements which require high computational costs, are the available solutions to tackle locking issues. The present 16-DOFs IGA is found to be efficient because of possessing a good rate of convergence and providing locking-free stable responses for micromorphic shells. Such a conclusion is found from several comparative studies with available data in the well-known macro-scale benchmark problems based on the classical elasticity as well as the corresponding numerical examples studied in nano-scale beam-, plate-, cylindrical shell- and spherical shell-type structures on the basis of the micromorphic continuum theory.

scholarly journals Multigrid solvers for immersed finite element methods and immersed isogeometric analysis

2019 ◽  
Vol 65 (3) ◽  
pp. 807-838 ◽  
Author(s):  
F. de Prenter ◽  
C. V. Verhoosel ◽  
E. H. van Brummelen ◽  
J. A. Evans ◽  
C. Messe ◽  
...  
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Isogeometric Analysis ◽  
Convergence Rates ◽  
Computational Cost ◽  
Iterative Solvers ◽  
System Matrix ◽  
B Splines ◽  
Immersed Finite Element ◽  
Immersed Finite Element Methods

AbstractIll-conditioning of the system matrix is a well-known complication in immersed finite element methods and trimmed isogeometric analysis. Elements with small intersections with the physical domain yield problematic eigenvalues in the system matrix, which generally degrades efficiency and robustness of iterative solvers. In this contribution we investigate the spectral properties of immersed finite element systems treated by Schwarz-type methods, to establish the suitability of these as smoothers in a multigrid method. Based on this investigation we develop a geometric multigrid preconditioner for immersed finite element methods, which provides mesh-independent and cut-element-independent convergence rates. This preconditioning technique is applicable to higher-order discretizations, and enables solving large-scale immersed systems at a computational cost that scales linearly with the number of degrees of freedom. The performance of the preconditioner is demonstrated for conventional Lagrange basis functions and for isogeometric discretizations with both uniform B-splines and locally refined approximations based on truncated hierarchical B-splines.

An enriched formulation of isogeometric analysis applied to the dynamical response of bars and trusses

2020 ◽  
Vol 37 (7) ◽  
pp. 2439-2466
Author(s):  
Mateus Rauen ◽  
Roberto Dalledone Machado ◽  
Marcos Arndt
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Isogeometric Analysis ◽  
Computer Aided Design ◽  
Numerical Solutions ◽  
Dynamical Response ◽  
Generalized Finite Element Method ◽  
Content Type ◽  
High Convergence Rate ◽  
Element Method

Purpose This study aims to present a new hybrid formulation based on non-uniform rational b-splines functions and enrichment strategies applied to free and forced vibration of straight bars and trusses. Design/methodology/approach Based on the idea of enrichment from generalized finite element method (GFEM)/extended finite element method (XFEM), an extended isogeometric formulation (partition of unity isogeometric analysis [PUIGA]) is conceived. By numerical examples the methods are tested and compared with isogeometric analysis, finite element method and GFEM in terms of convergence, error spectrum, conditioning and adaptivity capacity. Findings The results show a high convergence rate and accuracy for PUIGA and the advantage of input enrichment functions and material parameters on parametric space. Originality/value The enrichment strategies demonstrated considerable improvements in numerical solutions. The applications of computer-aided design mapped enrichments applied to structural dynamics are not known in the literature.

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