Josef Kiendl

A natural framework for isogeometric fluid-structure interaction based on BEM-shell coupling

Abstract The interaction between thin structures and incompressible Newtonian fluids is ubiquitous both in nature and in industrial applications. In this paper we present an isogeometric formulation of such problems which exploits a boundary integral formulation of Stokes equations to model the surrounding flow, and a non linear Kirchhoff–Love shell theory to model the elastic behavior of the structure. We propose three different coupling strategies: a monolithic, fully implicit coupling, a staggered, elasticity driven coupling, and a novel semi-implicit coupling, where the effect of the surrounding flow is incorporated in the non-linear terms of the solid solver through its damping characteristics. The novel semi-implicit approach is then used to demonstrate the power and robustness of our method, which fits ideally in the isogeometric paradigm, by exploiting only the boundary representation (B-Rep) of the thin structure middle surface.

Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro- and nano-beam models

As a first step, variational formulations and governing equations with boundary conditions are derived for a pair of Euler–Bernoulli beam bending models following a simplified version of Mindlin’s strain gradient elasticity theory of form II. For both models, this leads to sixth-order boundary value problems with new types of boundary conditions that are given additional attributes singly and doubly, referring to a physically relevant distinguishing feature between free and prescribed curvature, respectively. Second, the variational formulations are analyzed with rigorous mathematical tools: the existence and uniqueness of weak solutions are established by proving continuity and ellipticity of the associated symmetric bilinear forms. This guarantees optimal convergence for conforming Galerkin discretization methods. Third, the variational analysis is extended to cover two other generalized beam models: another modification of the strain gradient elasticity theory and a modified version of the couple stress theory. A model comparison reveals essential differences and similarities in the physicality of these four closely related beam models: they demonstrate essentially two different kinds of parameter-dependent stiffening behavior, where one of these kinds (possessed by three models out of four) provides results in a very good agreement with the size effects of experimental tests. Finally, numerical results for isogeometric Galerkin discretizations with B-splines confirm the theoretical stability and convergence results. Influences of the gradient and thickness parameters connected to size effects, boundary layers and dispersion relations are studied thoroughly with a series of benchmark problems for statics and free vibrations. The size-dependency of the effective Young’s modulus is demonstrated for an auxetic cellular metamaterial ruled by bending-dominated deformation of cell struts.

Arbitrary-degree T-splines for isogeometric analysis of fully nonlinear Kirchhoff–Love shells ☆

Abstract This paper focuses on the employment of analysis-suitable T-spline surfaces of arbitrary degree for performing structural analysis of fully nonlinear thin shells. Our aim is to bring closer a seamless and flexible integration of design and analysis for shell structures. The local refinement capability of T-splines together with the Kirchhoff–Love shell discretization, which does not use rotational degrees of freedom, leads to a highly efficient and accurate formulation. Trimmed NURBS surfaces, which are ubiquitous in CAD programs, cannot be directly applied in analysis, however, T-splines can reparameterize these surfaces leading to analysis-suitable untrimmed T-spline representations. We consider various classical nonlinear benchmark problems where the cylindrical and spherical geometries are exactly represented and point loads are accurately captured through local h -refinement. Taking advantage of the higher inter-element continuity of T-splines, smooth stress resultants are plotted without using projection methods. Finally, we construct various trimmed NURBS surfaces with Rhino, an industrial and general-purpose CAD program, convert them to T-spline surfaces, and directly use them in analysis.

An anisotropic constitutive model for immersogeometric fluid–structure interaction analysis of bioprosthetic heart valves

This paper considers an anisotropic hyperelastic soft tissue model, originally proposed for native valve tissue and referred to herein as the Lee-Sacks model, in an isogeometric thin shell analysis framework that can be readily combined with immersogeometric fluid-structure interaction (FSI) analysis for high-fidelity simulations of bioprosthetic heart valves (BHVs) interacting with blood flow. We find that the Lee-Sacks model is well-suited to reproduce the anisotropic stress-strain behavior of the cross-linked bovine pericardial tissues that are commonly used in BHVs. An automated procedure for parameter selection leads to an instance of the Lee-Sacks model that matches biaxial stress-strain data from the literature more closely, over a wider range of strains, than other soft tissue models. The relative simplicity of the Lee-Sacks model is attractive for computationally-demanding applications such as FSI analysis and we use the model to demonstrate how the presence and direction of material anisotropy affect the FSI dynamics of BHV leaflets.

Isogeometric phase-field modeling of brittle and ductile fracture in shell structures

Phase-field modeling of brittle and ductile fracture is a modern promising approach that enables a unified description of complicated failure processes (including crack initiation, propagation, branching, merging), as well as its efficient numerical treatment [1-4]. In the present work, we apply this approach to model fracture in shell structures, considering both thin and thick shells. For thin shells, we use an isogeometric Kirchhoff-Love shell formulation [5-6], which exploits the high continuity of the isogeometric shape functions in order to avoid rotational degrees of freedom, i.e., the shell geometry is modeled as a surface and its deformation is fully described by the displacements of this surface. For thick shells, we use an isogeometric assumed natural strain (ANS) solid shell formulation [7], i.e., a 3D solid formulation enhanced with the ANS method in order to alleviate geometrical locking effects. According to the discretization of the structural formulations, an isogeometric basis is also used for the phase-field. While the phase-field fracture formulation for solid shells is basically the same as for standard solids, some reformulation is necessary for thin shells, accounting for the interaction of stresses devoted to membrane and bending deformation. We test both formulations on several numerical examples and perform comparisons of the results obtained by the two methods to each other as well as to reference solutions, which confirm the validity and applicability of the presented methods.

ISOGEOMETRIC GALERKIN METHODS FOR GRADIENT-ELASTIC BARS, BEAMS, MEMBRANES AND PLATES

Isogeometric Galerkin methods are used to analyse plate and beam bending problems as well as membrane and bar models based on Mindlin’s strain gradient elasticity theory for generalized continua. The current strain gradient models include higher-order displacement gradients combined with length scale parameters enriching the strain and kinetic energies of the classical elasticity and hence resulting in higher-order partial differential equations with corresponding non-standard boundary conditions. The problems are first formulated within appropriate higher-order Sobolev space settings and then discretized by utilizing Galerkin methods with isogeometric NURBS basis functions providing appropriate higher-order continuity properties. Example benchmark problems illustrate the convergence properties of the methods.