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We present an adaptive space-time phase field formulation for dynamic fracture of brittle shells. Their deformation is characterized by the Kirchhoff–Love thin shell theory using a curvilinear surface description. All kinematical objects are defined on the shell’s mid-plane. The evolution equation for the phase field is determined by the minimization of an energy functional based on Griffith’s theory of brittle fracture. Membrane and bending contributions to the fracture process are modeled separately and a thickness integration is established for the latter. The coupled system consists of two nonlinear fourth-order PDEs and all quantities are defined on an evolving two-dimensional manifold. Since the weak form requires C 1 -continuity, isogeometric shape functions are used. The mesh is adaptively refined based on the phase field using Locally Refinable (LR) NURBS. Time is discretized based on a generalized- α method using adaptive time-stepping, and the discretized coupled system is solved with a monolithic Newton–Raphson scheme. The interaction between surface deformation and crack evolution is demonstrated by several numerical examples showing dynamic crack propagation and branching.

This paper builds on a recently developed immersogeometric fluid–structure interaction (FSI) methodology for bioprosthetic heart valve (BHV) modeling and simulation. It enhances the proposed framework in the areas of geometry design and constitutive modeling. With these enhancements, BHV FSI simulations may be performed with greater levels of automation, robustness and physical realism. In addition, the paper presents a comparison between FSI analysis and standalone structural dynamics simulation driven by prescribed transvalvular pressure, the latter being a more common modeling choice for this class of problems. The FSI computation achieved better physiological realism in predicting the valve leaflet deformation than its standalone structural dynamics counterpart.

We develop and analyze an ultraweak variational formulation for a variant of the Kirchhoff–Love plate bending model. Based on this formulation, we introduce a discretization of the discontinuous Petrov–Galerkin type with optimal test functions (DPG). We prove well-posedness of the ultraweak formulation and quasi-optimal convergence of the DPG scheme. The variational formulation and its analysis require tools that control traces and jumps in H^2 (standard Sobolev space of scalar functions) and H(\\operatorname {div}\\,\\, \\mathbf{div}\\!) (symmetric tensor functions with L_2-components whose twice iterated divergence is in L_2), and their dualities. These tools are developed in two and three spatial dimensions. One specific result concerns localized traces in a dense subspace of H(\\operatorname {div}\\,\\, \\mathbf{div}\\!). They are essential to construct basis functions for an approximation of H(\\operatorname {div}\\,\\, \\mathbf{div}\\!). To illustrate the theory we construct basis functions of the lowest order and perform numerical experiments for a smooth and a singular model solution. They confirm the expected convergence behavior of the DPG method both for uniform and adaptively refined meshes.

A frequency error estimation is presented for the isogeometric free vibration analysis of Kirchhoff-Love cylindrical shells using both quadratic and cubic basis functions. By analyzing the discrete isogeometric equations with the aid of harmonic wave assumption, the frequency error measures are rationally derived for the quadratic and cubic formulations for Kirchhoff-Love cylindrical shells. In particular, the governing relationship of the continuum frequency for Kirchhoff-Love cylindrical shells is naturally embedded into the frequency error measures without the need of explicit frequency expressions, which usually are not trivial for the shell problems. In accordance with these theoretical findings, the 2nd and 4th orders of frequency accuracy are attained for the isogeometric schemes using quadratic and cubic basis functions, respectively. Numerical results not only thoroughly verify the theoretical convergence rates of frequency solutions, but also manifest an excellent magnitude match between numerical and theoretical frequency errors for the isogeometric free vibration analysis of Kirchhoff-Love cylindrical shells.

Biomembranes play a central role in various phenomena like locomotion of cells, cell-cell interactions, packaging and transport of nutrients, transmission of nerve impulses, and in maintaining organelle morphology and functionality. During these processes, the membranes undergo significant morphological changes through deformation, scission, and fusion. Modelling the underlying mechanics of such morphological changes has traditionally relied on reduced order axisymmetric representations of membrane geometry and deformation. Axisymmetric representations, while robust and extensively deployed, suffer from their inability to model-symmetry breaking deformations and structural bifurcations. To address this limitation, a three-dimensional computational mechanics framework for high fidelity modelling of biomembrane deformation is presented. The proposed framework brings together Kirchhoff–Love thin-shell kinematics, Helfrich-energy-based mechanics, and state-of-the-art numerical techniques for modelling deformation of surface geometries. Lipid bilayers are represented as spline-based surface discretizations immersed in a three-dimensional space; this enables modelling of a wide spectrum of membrane geometries, boundary conditions, and deformations that are physically admissible in a three-dimensional space. The mathematical basis of the framework and its numerical machinery are presented, and their utility is demonstrated by modelling three classical, yet non-trivial, membrane deformation problems: formation of tubular shapes and their lateral constriction, Piezo1-induced membrane footprint generation and gating response, and the budding of membranes by protein coats during endocytosis. For each problem, the full three-dimensional membrane deformation is captured, potential symmetry-breaking deformation paths identified, and various case studies of boundary and load conditions are presented. Using the endocytic vesicle budding as a case study, we also present a ‘phase diagram’ for its symmetric and broken-symmetry states.

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.

We develop low-order triangular virtual elements for linear Kirchhoff–Love shells from an engineering point of view. Flat element geometry is considered, which enables a direct shell discretization with no need for a curvilinear coordinate system or predefined initial mapping. Along with the assumed linearity of the problem, the superposition of the uncoupled membrane and plate energies is performed by unifying aspects of the virtual element method when applied to linear two-dimensional elasticity and plate bending problems. We explore low-order cases, namely linear to quadratic membrane displacements and quadratic to cubic deflection polynomial approximations such that no internal degrees of freedom are needed. For all elements, a single stabilization available in the literature is employed to stabilize the element formulations. Numerical examples of static problems show that the presented formulation is capable of solving complex shell problems. Possible extensions are discussed in future works.

This work focuses on the study of several computational challenges arising when trimmed surfaces are directly employed for the isogeometric analysis of Kirchhoff–Love shells. To cope with these issues and to resolve mechanical and/or geometrical features of interest, we exploit the local refinement capabilities of hierarchical B-splines. In particular, we show numerically that local refinement is suited to effectively impose Dirichlet-type boundary conditions in a weak sense, where this easily allows to overcome the issue of over-constraining of trimmed elements. Moreover, we highlight how refinement can alleviate the spurious coupling stemming from disjoint supports of basis functions in the presence of “small” trimmed geometrical features such as thin holes. These phenomena are particularly pronounced in surface models defined by complex trimming patterns and with details at different scales. In this contribution we focus our effort on the analysis of single-patch geometries, where we show through several numerical examples the benefits and computational efficiency of the proposed methodology.

We derive a hyperelastic shell formulation based on the Kirchhoff–Love shell theory and isogeometric discretization, where we take into account the out-of-plane deformation mapping. Accounting for that mapping affects the curvature term. It also affects the accuracy in calculating the deformed-configuration out-of-plane position, and consequently the nonlinear response of the material. In fluid–structure interaction analysis, when the fluid is inside a shell structure, the shell midsurface is what it would know. We also propose, as an alternative, shifting the “midsurface” location in the shell analysis to the inner surface, which is the surface that the fluid should really see. Furthermore, in performing the integrations over the undeformed configuration, we take into account the curvature effects, and consequently integration volume does not change as we shift the “midsurface” location. We present test computations with pressurized cylindrical and spherical shells, with Neo-Hookean and Fung’s models, for the compressible- and incompressible-material cases, and for two different locations of the “midsurface.” We also present test computation with a pressurized Y-shaped tube, intended to be a simplified artery model and serving as an example of cases with somewhat more complex geometry.

This paper presents a new triangular nonlinear shell finite element with a novel kinematic model suitable for simulation with large displacements and rotations, herein introduced as “T6-3iKL”. This element has 6 nodes, a quadratic displacement field, and a linear rotation field based on Rodrigues incremental rotation parameters, giving in total 21 degrees of freedom. The novelty of this shell element concerns a new kinematic model with properties from Kirchhoff-Love shell theory, making it possible to eliminate the drilling DOF in the rotation field (compared to Mindlin-Reissner models), approximating the rotation continuity between adjacent elements by a single scalar and allowing multiple branch connections in the mesh, making this element very simple and interesting, with no artificial parameters imputed by the code (such as penalties or Lagrange multipliers). The element permits the implementation of different material constitutive equations, including elastic anisotropic models, and the thickness of the shell is optionally allowed to change through the simulation. The model developed in this article is numerically implemented and the results compared to different references in multiple examples, showing the consistency and reliability of the new formulation. It is believed that this new versatile triangular shell element, with no necessity of artificial penalty calibration, simple kinematics, a relatively small number of DOFs, geometric exactness, the possibility to use 3D material constitutive models, and easy connection with multiple branched shells and beams, implemented together with reliable mesh generation, may be an effective option for shell simulation in many engineering applications.