Fehmi Cirak

Subdivision surfaces: a new paradigm for thin-shell finite-element analysis

We develop a new paradigm for thin-shell finite-element analysis based on the use of subdivision surfaces for (i) describing the geometry of the shell in its undeformed configuration, and (ii) generating smooth interpolated displacement fields possessing bounded energy within the strict framework of the Kirchhoff–Love theory of thin shells. The particular subdivision strategy adopted here is Loops scheme, with extensions such as required to account for creases and displacement boundary conditions. The displacement fields obtained by subdivision are H2 and, consequently, have a finite Kirchhoff–Love energy. The resulting finite elements contain three nodes and element integrals are computed by a one-point quadrature. The displacement field of the shell is interpolated from nodal displacements only. In particular, no nodal rotations are used in the interpolation. The interpolation scheme induced by subdivision is non-local, i.e. the displacement field over one element depend on the nodal displacements of the element nodes and all nodes of immediately neighbouring elements. However, the use of subdivision surfaces ensures that all the local displacement fields thus constructed combine conformingly to define one single limit surface. Numerical tests, including the Belytschko et al. [10] obstacle course of benchmark problems, demonstrate the high accuracy and optimal convergence of the method.

Integrated modeling, finite-element analysis, and engineering design for thin-shell structures using subdivision

Many engineering design applications require geometric modeling and mechanical simulation of thin flexible structures, such as those found in the automotive and aerospace industries. Traditionally, geometric modeling, mechanical simulation, and engineering design are treated as separate modules requiring different methods and representations. Due to the incompatibility of the involved representations the transition from geometric modeling to mechanical simulation, as well as in the opposite direction, requires substantial effort. However, for engineering design purposes efficient transition between geometric modeling and mechanical simulation is essential. We propose the use of subdivision surfaces as a common foundation for modeling, simulation, and design in a unified framework. Subdivision surfaces provide a flexible and efficient tool for arbitrary topology free-form surface modeling, avoiding many of the problems inherent in traditional spline patch based approaches. The underlying basis functions are also ideally suited for a finite-element treatment of the so-called thin-shell equations, which describe the mechanical behavior of the modeled structures. The resulting solvers are highly scalable, providing an efficient computational foundation for design exploration and optimization. We demonstrate our claims with several design examples, showing the versatility and high accuracy of the proposed method.

A posteriori error estimation and adaptivity for linear elasticity using the reciprocal theorem

The reciprocal theorem of Betti and Rayleigh or in other notions duality arguments are used to derive error estimators for the finite element approximation of various quantities, including local variables like single displacements and stresses. The proposed error estimator is evaluated solving the set of equations for an additional right-hand side, simply applying the energy norm error estimators two times. Furthermore, a general h-adaptive algorithm is introduced which allows us to optimize meshes with respect to different user specified variables. The efficiency of the current approach is demonstrated for plate and shell examples.

A posteriori error estimation and adaptivity for elastoplasticity using the reciprocal theorem

We present a-posteriori error estimators and adaptive methods for the finite element approximation of nonlinear problems and especially elastoplasticity. The main characteristic of the proposed method is the introduction of duality techniques or in other notions the reciprocal theorem. For error estimation at an equilibrium point the nonlinear boundary value problem and an additional linearized dual problem are considered. The loading of the dual problem is specifically designed for capturing the influence of the errors of the entire domain to the considered variable. Our approach leads to easy computable refinement indicators for locally or integrally defined variables. For instationary problems as elastoplasticity, in a first step, we neglect the errors due to time discretization, and evaluate the error indicators within each time step for a stationary problem. The versatility of the presented framework is demonstrated with numerical examples.

Boundary element based multiresolution shape optimisation in electrostatics

We consider the shape optimisation of high-voltage devices subject to electrostatic field equations by combining fast boundary elements with multiresolution subdivision surfaces. The geometry of the domain is described with subdivision surfaces and different resolutions of the same geometry are used for optimisation and analysis. The primal and adjoint problems are discretised with the boundary element method using a sufficiently fine control mesh. For shape optimisation the geometry is updated starting from the coarsest control mesh with increasingly finer control meshes. The multiresolution approach effectively prevents the appearance of non-physical geometry oscillations in the optimised shapes. Moreover, there is no need for mesh regeneration or smoothing during the optimisation due to the absence of a volume mesh. We present several numerical experiments and one industrial application to demonstrate the robustness and versatility of the developed approach.

Shape optimisation with multiresolution subdivision surfaces and immersed finite elements

Abstract We develop a new optimisation technique that combines multiresolution subdivision surfaces for boundary description with immersed finite elements for the discretisation of the primal and adjoint problems of optimisation. Similar to wavelets, multiresolution surfaces represent the domain boundary using a coarse control mesh and a sequence of detail vectors. Based on the multiresolution decomposition efficient and fast algorithms are available for reconstructing control meshes of varying fineness. During shape optimisation the vertex coordinates of control meshes are updated using the computed shape gradient information. By virtue of the multiresolution editing semantics, updating the coarse control mesh vertex coordinates leads to large-scale geometry changes and, conversely, updating the fine control mesh coordinates leads to small-scale geometry changes. In our computations we start by optimising the coarsest control mesh and refine it each time the cost function reaches a minimum. This approach effectively prevents the appearance of non-physical boundary geometry oscillations and control mesh pathologies, like inverted elements. Independent of the fineness of the control mesh used for optimisation, on the immersed finite element grid the domain boundary is always represented with a relatively fine control mesh of fixed resolution. With the immersed finite element method there is no need to maintain an analysis suitable domain mesh. In some of the presented two and three-dimensional elasticity examples the topology derivative is used for introducing new holes inside the domain. The merging or removing of holes is not considered.

Generic programming techniques for parallelizing and extending procedural finite element programs

We outline an approach for extending procedural finite-element software components using generic programming. A layer of generic software components consisting of C++ containers and algorithms is used for parallelization of the finite-element solver and for solver coupling in multi-physics applications. The advantages of generic programming in connection with finite-element codes are discussed and compared with those of object-oriented programming. The use of the proposed generic programming techniques is demonstrated in a tutorial fashion through basic illustrative examples as well as code excerpts from a large-scale finite-element program for serial and parallel computing platforms.

A virtual test facility for simulating detonation-induced fracture of thin flexible shells

The fluid-structure interaction simulation of detonation- and shock-wave-loaded fracturing thin-walled structures requires numerical methods that can cope with large deformations as well as topology changes. We present a robust level-set-based approach that integrates a Lagrangian thin shell finite element solver with fracture and fragmentation capabilities with an Eulerian Cartesian detonation solver with optional dynamic mesh adaptation. As an application example, the rupture of a thin aluminum tube due to the passage of an ethylene-oxygen detonation wave is presented.

Isogeometric analysis using manifold-based smooth basis functions

We present an isogeometric analysis technique that builds on manifold-based smooth basis functions for geometric modelling and analysis. Manifold-based surface construction techniques are well known in geometric modelling and a number of variants exist. Common to most is the concept of constructing a smooth surface by blending together overlapping patches (or, charts), as in differential geometry description of manifolds. Each patch on the surface has a corresponding planar patch with a smooth one-to-one mapping onto the surface. In our implementation, manifold techniques are combined with conformal parameterisations and the partition-of-unity method for deriving smooth basis functions on unstructured quadrilateral meshes. Each vertex and its adjacent elements on the surface control mesh have a corresponding planar patch of elements. The star-shaped planar patch with congruent wedge-shaped elements is smoothly parameterised with copies of a conformally mapped unit square. The conformal maps can be easily inverted in order to compute the transition functions between the different planar patches that have an overlap on the surface. On the collection of star-shaped planar patches the partition of unity method is used for approximation. The smooth partition of unity, or blending functions, are assembled from tensor-product b-spline segments defined on a unit square. On each patch a polynomial with a prescribed degree is used as a local approximant. In order to obtain a mesh-based approximation scheme the coefficients of the local approximants are expressed in dependence of vertex coefficients. This yields a basis function for each vertex of the mesh which is smooth and non-zero over a vertex and its adjacent elements. Our numerical simulations indicate the optimal convergence of the resulting approximation scheme for Poisson problems and near optimal convergence for thin-plate and thin-shell problems discretised with structured and unstructured quadrilateral meshes.

Average Corotation of Line Segments Near a Point and Vortex Identification

An easy-to-interpret kinematic quantity measuring the average corotation of material line segments near a point is introduced and applied to vortex identification. At a given point, the vector of average corotation of line segments is defined as the average of the instantaneous local rigid-body rotation over “all planar cross sections” passing through the examined point. The vortex-identification method based on average corotation is a one-parameter, region-type local method sensitive to the axial stretching rate as well as to the inner configuration of the velocity gradient tensor. The method is derived from a well-defined interpretation of the local flow kinematics to determine the “plane of swirling” and is also applicable to compressible and variable-density flows. Practical application to direct numerical simulation datasets includes a hairpin vortex of boundary-layer transition, the reconnection process of two Burgers vortices, a flow around an inclined flat plate, and a flow around a revolving inse...