Subdivision based mixed methods for isogeometric analysis of linear and nonlinear nearly incompressible materials

Summary We develop a mixed formulation for incompressible hyperelastodynamics based on a continuum modeling framework recently developed in the work of Liu and Marsden and smooth generalizations of the Taylor‐Hood element based on nonuniform rational B‐splines (NURBSs). This continuum formulation draws a link between computational fluid dynamics and computational solid dynamics. This link inspires an energy stability estimate for the spatial discretization, which favorably distinguishes the formulation from the conventional mixed formulations for finite elasticity. The inf‐sup condition is utilized to provide a bound for the pressure field. The generalized‐α method is applied for temporal discretization, and a nested block preconditioner is invoked for the solution procedure. The inf‐sup stability for different pairs of NURBS elements is elucidated through numerical assessment. The convergence rate of the proposed formulation with various combinations of mixed elements is examined by the manufactured solution method. The numerical scheme is also examined under compressive and tensile loads for isotropic and anisotropic hyperelastic materials. Finally, a suite of dynamic problems is numerically studied to corroborate the stability and conservation properties.

Section: Overview Of the Proposed Methodsmentioning

confidence: 99%

An energy‐stable mixed formulation for isogeometric analysis of incompressible hyperelastodynamics

Marsden

Tao

2019

“…() Numerous methods have been proposed to satisfy this condition in the context of the finite element method, the most popular to this point being the Taylor‐Hood family with quadratic interpolations for the displacement (or velocity) field. It was observed in the works of Kadapa et al and Rüberg and Cirak that NURBS combinations Q a k /Q( a ‐1) equivalent to the Taylor‐Hood family are unstable in the context of k ‐refinement. To address this issue, stable subdivision‐based mixed methods were proposed in the works of Rüberg and Cirak() for fluid mechanics and fluid structure interactions and were further studied in other works.…”

Section: Introductionmentioning

confidence: 99%

“…To address this issue, stable subdivision‐based mixed methods were proposed in the works of Rüberg and Cirak() for fluid mechanics and fluid structure interactions and were further studied in other works. () In addition, subdivision‐based mixed methods have been recently adopted in the work of Kadapa et al for quasi‐incompressible elasticity. In the case of elasticity, the main reason for the use of mixed elements is to overcome locking in the incompressible limit since NURBS shape functions suffer from the same locking issues as Lagrange shape functions .…”

Section: Introductionmentioning

confidence: 99%

Mixed isogeometric analysis of strongly coupled diffusion in porous materials

Dortdivanlioglu

Krischok

Veiga

et al. 2018

Summary In this paper, we develop a mixed isogeometric analysis approach based on subdivision stabilization to study strongly coupled diffusion in solids in both small and large deformation ranges. Coupling the fluid pressure and the solid deformation, the mixed formulation suffers from numerical instabilities in the incompressible and the nearly incompressible limit due to the violation of the inf‐sup condition. We investigate this issue using subdivision‐stabilized nonuniform rational B‐spline (NURBS) elements, as well as different families of mixed isogeometric analysis techniques, and assess their stability through a numerical inf‐sup test. Furthermore, the validity of the inf‐sup stability test in poromechanics is supported by a mathematical proof concerning the corresponding stability estimate. Finally, two numerical examples involving a rigid strip foundation on saturated soil and a swelling hydrogel structure are presented to validate the stability and to demonstrate the robustness of the proposed approach.

“…Comprehensive research and development of NURBS-based FEMs has been undertaken and widely published by Hughes and co-workers (see, for instance, [1,2] and references therein). Such strategies are today commonly referred to as Isogeometric Analysis (IGA), and their applications include shell analysis, nearly incompressible solid mechanics, structural vibrations, structural optimisation, phase-transition phenomena, turbulent fluid dynamics and fluid-structure interaction (see, for instance, [3][4][5][6][7][8][9][10]). The primary motivation behind the development of IGA are the direct utilisation of geometry models employed in computer-aided design and the ease of constructing and implementing higher order basis functions.…”

Section: Introductionmentioning

confidence: 99%

“…Many of the strategies developed in the context of Galerkin finite element methods (Galerkin-FEM) with Lagrangian basis functions have been reformulated in the isogeometric setting. This includes stabilisation techniques for advection dominated problems in fluid mechanics as well as strategies to circumvent or satisfy the Ladyzhenskaya-BabuL ska-Brezzi (LBB) condition in incompressible fluid or solid materials (see, for instance, [2,4,5,11,12]).…”

Section: Introductionmentioning

confidence: 99%

NURBS based least‐squares finite element methods for fluid and solid mechanics

Kadapa

Dettmer

Perić

2015

Self Cite

This contribution investigates the performance of a least-squares finite element method based on nonuniform rational B-splines (NURBS) basis functions. The least-squares functional is formulated directly in terms of the strong form of the governing equations and boundary conditions. Thus, the introduction of auxiliary variables is avoided, but the order of the basis functions must be higher or equal to the order of the highest spatial derivatives. The methodology is applied to the incompressible Navier-Stokes equations and to linear as well as nonlinear elastic solid mechanics. The numerical examples presented feature convective effects and incompressible or nearly incompressible material. The numerical results, which are obtained with equal-order interpolation and without any stabilisation techniques, are smooth and accurate. It is shown that for p and h refinement, the theoretical rates of convergence are achieved.If higher order problems are not rewritten as first order equation systems by means of auxiliary solution variables but, instead, sufficiently smooth approximation spaces are employed, then the conditioning of the system matrix deteriorates [17]. This may adversely affect practical computations.This work is organised as follows. In Sections 2 and 3, the fundamentals of LSFEM and of IGA and NURBS are revised, respectively. NURBS-based LSFEM is applied to incompressible Newtonian fluid flow in Section 4. In Section 5, the methodology is extended to linear and nonlinear elasticity. Sections 4 and 5 also include the presentation of a number of numerical examples. Conclusions are summarised in Section 5. LEAST-SQUARES FINITE ELEMENT METHODS Basic least-squares finite element method methodologyThis section, which closely follows [18], describes the fundamental idea behind LSFEM.Let be a bounded domain in R d ; d D 1; 2 or 3, with a boundary . Let L be a linear differential operator and R be a boundary operator. Consider the problem given by Figure 3. Kovasznay flow: velocity and pressure contour plots for 20 20 mesh with Q 2 and Q 3 NURBS.where D D [ N is the boundary of the body . The vector fields u; g; f ; t and n represent, respectively, the displacement, the prescribed displacement on D , the body force, the prescribed traction on N and the outward unit normal on . The stress tensor is defined as D 2 " C tr."/ ;