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Computational modeling of thin biological membranes can aid the design of better medical devices. Remarkable biological membranes include skin, alveoli, blood vessels, and heart valves. Isogeometric analysis is ideally suited for biological membranes since it inherently satisfies the C1-requirement for Kirchhoff-Love kinematics. Yet, current isogeometric shell formulations are mainly focused on linear isotropic materials, while biological tissues are characterized by a nonlinear anisotropic stress-strain response. Here we present a thin shell formulation for thin biological membranes. We derive the equilibrium equations using curvilinear convective coordinates on NURBS tensor product surface patches. We linearize the weak form of the generic linear momentum balance without a particular choice of a constitutive law. We then incorporate the constitutive equations that have been designed specifically for collagenous tissues. We explore three common anisotropic material models: Mooney-Rivlin, May Newmann-Yin, and Gasser-Ogden-Holzapfel. Our work will allow scientists in biomechanics and mechanobiology to adopt the constitutive equations that have been developed for solid three-dimensional soft tissues within the framework of isogeometric thin shell analysis.

Universal meshes have recently appeared in the literature as a computationally efficient and robust paradigm for the generation of conforming simplicial meshes for domains with evolving boundaries. The main idea behind a universal mesh is to immerse the moving boundary in a background mesh (the universal mesh), and to produce a mesh that conforms to the moving boundary at any given time by adjusting a few of elements of the background mesh. In this manuscript we present the application of universal meshes to the simulation of brittle fracturing. To this extent, we provide a high level description of a crack propagation algorithm and showcase its capabilities. Alongside universal meshes for the simulation of brittle fracture, we provide other examples for which universal meshes prove to be a powerful tool, namely fluid flow past moving obstacles. Lastly, we conclude the manuscript with some remarks on the current state of universal meshes and future directions.Dedicated to Michael Ortiz on the occasion of his 60 th birthday.

We propose a family of methods for simulating two-dimensional incompressible, low Reynolds number flow around a moving obstacle whose motion is prescribed. The methods make use of a universal mesh: a fixed background mesh that adapts to the geometry of the immersed obstacle at all times by adjusting a few elements in the neighborhood of the obstacle's boundary. The resulting mesh provides a conforming triangulation of the fluid domain over which discretizations of any desired order of accuracy in space and time can be constructed using standard finite element spaces together with off-the-shelf time integrators. We demonstrate the approach by using Taylor-Hood elements to approximate the fluid velocity and pressure. To integrate in time, we consider implicit Runge-Kutta schemes as well as a fractional step scheme. We illustrate the methods and study their convergence numerically via examples that involve flow around obstacles that undergo prescribed deformations.Organization. This paper is organized as follows. In Section 2, we recall the governing equations for incompressible, viscous flow around a moving obstacle with prescribed evolution, and we recast the equations in weak form. In Section 3, we propose a discretization of the aforementioned equations using a universal mesh in conjunction with Taylor-Hood finite elements [61]. To integrate in time, we propose the use of implicit Runge-Kutta schemes as well as a fractional step scheme. In Section 4, we apply the proposed methods to simulate flow around various obstacles with prescribed evolution: The setup we have described thus far offers the freedom to employ a time integrator of one's choosing to numerically integrate (17), a system of DAEs of index 2, from t D t n 1 to t D t n . We present two examples of integration schemes: a singly diagonally implicit Runge-Kutta (SDIRK) scheme [69,70] and a fractional step scheme [71][72][73]. In accordance with common guidelines for numerically solving DAEs, the SDIRK schemes we consider are stiffly accurate (and hence L-stable) methods [70,74]. The same schemes are considered by, for instance, [75,76] in their studies of high-order methods for the Navier-Stokes equations on fixed domains.We immersed the oscillating disk in a domain D D OE 6; 6 OE 3; 3 and prescribed boundary conditions ‡ In the case of the fractional step scheme, the error in p was measured at t D T t=2 rather than at t D T .

We introduce a method to mesh the boundary of a smooth, open domain in R 3 immersed in a mesh of tetrahedra. The mesh follows by mapping a specific collection of triangular faces in the mesh to . Two types of surface meshes follow: (a) a mesh that exactly meshes , and (b) meshes that approximate to any order, by interpolating the map over the selected faces; that is, an isoparametric approximation to . The map we use to deform the faces is the closest point projection to . We formulate conditions for the closest point projection to define a homeomorphism between each face and its image. These are conditions on some of the tetrahedra intersected by the boundary, and they essentially state that each such tetrahedron should (a) have a small enough diameter, and (b) have two of its dihedral angles be acute. We provide explicit upper bounds on the mesh size, and these can be computed on the fly.We showcase the quality of the resulting meshes with several numerical examples. More importantly, all surfaces in these examples were meshed with a single background mesh. This is an important feature for problems in which the geometry evolves or changes, because it could be possible for the background mesh to never change as the geometry does. In this case, the background mesh would be a universal mesh [1] for all these geometries. We expect the method introduced here to be the basis for the construction of universal meshes for domains in three dimensions. it defines then a triangulation with curved triangles that exactly lie on the surface. Additionally, and as we explain here, by constructing a finite element space over the set of positive faces, and then interpolating the map that defines the homeomorphism, we obtain an isoparametric approximation of the domain. In this way, the method could be used with standard finite element tools available in many commercial codes, and approximations of the geometry of any order can be constructed. For finite element spaces of the Lagrange type, this is tantamount to moving nodes on the positive faces to the surface.The motivation behind this method is that, as the geometry of the domain evolves, the same background mesh would be deformed to conform to the geometry of the domain for the entire (or for periods that do not depend on the mesh size or time step) of the simulation. Hence, no new vertices or faces are introduced in the background mesh as the geometry evolves. For finite element methods, this has the positive consequence of keeping intact the sparsity pattern of the matrix associated with the mesh as the domains evolves. For similar reasons, the method provides algorithmic advantages if iterating over the geometry of the domain is necessary, a feature that could be advantageous in explored in applications such as topology optimization or some fluid-structure interaction problems. It is important to note that we see the surface generation method presented here a step towards the method to generate tetrahedral meshes for deforming and moving domains from a fixed background mesh...

We describe an algorithm to recover a boundary-fitting triangulation for a bounded C 2 -regular domain Ω ⊂ R 3 immersed in a nonconforming background mesh of tetrahedra. The algorithm consists in identifying a polyhedral domain h bounded by facets in the background mesh and morphing h into a boundary-fitting polyhedral approximation Ω h of Ω. We discuss assumptions on the regularity of the domain, on element sizes and on specific angles in the background mesh that appear to render the algorithm robust. With the distinctive feature of involving just small perturbations of a few elements of the background mesh that are in the vicinity of the immersed boundary, the algorithm is designed to benefit numerical schemes for simulating free and moving boundary problems. In such problems, it is now possible to immerse an evolving geometry in the same background mesh, called a universal mesh, and recover conforming discretizations for it. In particular, the algorithm entirely avoids remeshing-type operations and its complexity scales approximately linearly with the number of elements in the vicinity of the immersed boundary. We include detailed examples examining its performance. KEYWORDSbackground meshes, evolving boundary, immersed boundary, mesh relaxation, moving mesh, 3D meshing 84

Summary We introduce a method to mesh the boundary Γ of a smooth, open domain in double-struckR3 immersed in a mesh of tetrahedra. The mesh follows by mapping a specific collection of triangular faces in the mesh to Γ. Two types of surface meshes follow: (a) a mesh that exactly meshes Γ, and (b) meshes that approximate Γ to any order, by interpolating the map over the selected faces; that is, an isoparametric approximation to Γ. The map we use to deform the faces is the closest point projection to Γ. We formulate conditions for the closest point projection to define a homeomorphism between each face and its image. These are conditions on some of the tetrahedra intersected by the boundary, and they essentially state that each such tetrahedron should (a) have a small enough diameter, and (b) have two of its dihedral angles be acute. We provide explicit upper bounds on the mesh size, and these can be computed on the fly. We showcase the quality of the resulting meshes with several numerical examples. More importantly, all surfaces in these examples were meshed with a single background mesh. This is an important feature for problems in which the geometry evolves or changes, because it could be possible for the background mesh to never change as the geometry does. In this case, the background mesh would be a universal mesh for all these geometries. We expect the method introduced here to be the basis for the construction of universal meshes for domains in three dimensions. Copyright © 2016 John Wiley & Sons, Ltd.

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