Petr Krysl

Meshless methods: An overview and recent developments

Meshless approximations based on moving least-squares, kernels, and partitions of unity are examined. It is shown that the three methods are in most cases identical except for the important fact that partitions of unity enable p-adaptivity to be achieved. Methods for constructing discontinuous approximations and approximations with discontinuous derivatives are also described. Next, several issues in implementation are reviewed: discretization (collocation and Galerkin), quadrature in Galerkin and fast ways of constructing consistent moving least-square approximations. The paper concludes with some sample calculations.

Analysis of thin shells by the element-free galerkin method

A meshless approach to the analysis of arbitrary Kirchhoff shells by the Element-Free Galerkin (EFG) method is presented. The shell theory used is geometrically exact and can be applied to deep shells. The method is based on moving least squares approximant. The method is meshless, which means that the discretization is independent of the geometric subdivision into “finite elements”. The satisfaction of the C1 continuity requirements is easily met by EFG since it requires only C1 weights; therefore, it is not necessary to resort to Mindlin-Reissner theory or to devices such as discrete Kirchhoff theory. The requirements of consistency are met by the use of a polynomial basis of quadratic or higher order. A subdivision similar to finite elements is used to provide a background mesh for numerical integration. The essential boundary conditions are enforced by Lagrange multipliers. Membrane locking, which is due to different approximation order for transverse and membrane displacements, is removed by using larger domains of influence with the quadratic basis, and by using quartic polynomial basis, which can prevent membrane locking completely. It is shown on the obstacle course for shells that the present technique performs well.

CHARMS: a simple framework for adaptive simulation

Finite element solvers are a basic component of simulation applications; they are common in computer graphics, engineering, and medical simulations. Although adaptive solvers can be of great value in reducing the often high computational cost of simulations they are not employed broadly. Indeed, building adaptive solvers can be a daunting task especially for 3D finite elements. In this paper we are introducing a new approach to produce conforming, hierarchical, adaptive refinement methods (CHARMS). The basic principle of our approach is to refine basis functions, not elements. This removes a number of implementation headaches associated with other approaches and is a general technique independent of domain dimension (here 2D and 3D), element type (e.g., triangle, quad, tetrahedron, hexahedron), and basis function order (piece-wise linear, higher order B-splines, Loop subdivision, etc.). The (un-)refinement algorithms are simple and require little in terms of data structure support. We demonstrate the versatility of our new approach through 2D and 3D examples, including medical applications and thin-shell animations.

The Element Free Galerkin method for dynamic propagation of arbitrary 3‐D cracks

A technique for modelling of arbitrary three-dimensional dynamically propagating cracks in elastic bodies by the Element-Free Galerkin (EFG) method with explicit time integration is described. The meshless character of this approach expedites the description of the evolving discrete model; in contrast to the finite element method no remeshing of the domain is required. The crack surface is defined by a set of triangular elements. Techniques for updating the surface description are reported. The paper concludes with several examples: a simulation of mixed-mode growth of a center crack, mode-I surface-breaking penny-shaped crack, penny-shaped crack growing under mixed-mode conditions in a cube, and a bar with centre through crack. Copyright

Deformation mechanism transitions in nanoscale fcc metals

We consider possible mechanisms that lead to transitions in the mechanisms of deformation in fcc metals and alloys. In particular, we propose that, when grain sizes are below a critical size (i.e. below 100 nm), deformation can occur via the emission of stacking faults from grain boundaries into the intragranular space. A model is developed that accounts for observed experimental data and which, in turn, shows how stacking-fault energy together with shear modulus determines achievable strength. A mechanism is proposed based on this model for transitions at both high and quasistatic strain rates, including grain-boundary sliding.

Finite element simulation of ring expansion and fragmentation: The capturing of length and time scales through cohesive models of fracture

The expanding ring test of Grady and Benson (1983) is taken as a convenient yet challenging validation problem for assessing the fidelity of cohesive models in situations involving ductile dynamical fracture. Attention has been restricted to 1100-0 aluminum samples. Fracture has been modelled by recourse to an irreversible cohesive law embedded into cohesive elements. The finite element model is three-dimensional and fully Lagrangian. In order to limit the extent of deformation-induced distortion, we resort to continuous adaptive remeshing. The cohesive behavior of the material is assumed to be rate independent and, consequently, all rate effects predicted by the calculations are due to inertia and the rate dependency in plastic deformation. The numerical simulations are revealed to be highly predictive of a number of observed features, including: the number of dominant and arrested necks; the fragmentation patterns; the dependence of the number of fragments and the fracture strain on the expansion speed; and the distribution of fragment sizes at fixed expansion speed.

Element-free Galerkin method : Convergence of the continuous and discontinuous shape functions

Abstract We consider numerical solutions of second-order elliptic partial differential equations, such as Laplaces equation, or linear elasticity, in two-dimensional, non-convex domains by the element-free Galerkin method (EFG). This is a meshless method in which the shape functions are constructed by using weight functions of compact support. For non-convex domains, two approaches to the determination of whether a node affects approximation at a particular point are used, a contained path criterion, and the visibility criterion. We show that for non-convex domains the visibility criterion leads to discontinuous weight functions and discontinuous shape functions. The resulting approximation is no longer conforming, and its convergence must be established by inspection of the so-called consistency term. We show that the variant of the element-free Galerkin method which uses the discontinuous shape functions, is convergent, and that, in the practically important case of linear shape functions, the convergence rate is not affected by the discontinuities. The convergence of the discontinuous approximation is first established by the classical and generalized patch test. As these tests do not provide an estimate of the convergence rate, the rate of convergence in the energy norm is examined, for both the continuous and discontinuous EFG shape functions and for smooth and non-smooth solutions by a direct inspection of the error terms.

Structure-preserving model reduction for mechanical systems

This paper focuses on methods of constructing of reduced-order models of mechanical systems which preserve the Lagrangian structure of the original system. These methods may be used in combination with standard spatial decomposition methods, such as the Karhunen–Loeve expansion, balancing, and wavelet decompositions. The model reduction procedure is implemented for three-dimensional finite-element models of elasticity, and we show that using the standard Newmark implicit integrator, significant savings are obtained in the computational costs of simulation. In particular simulation of the reduced model scales linearly in the number of degrees of freedom, and parallelizes well.

Anatomic geometry of sound transmission and reception in Cuvier's beaked whale (Ziphius cavirostris).

This study uses remote imaging technology to quantify, compare, and contrast the cephalic anatomy between a neonate female and a young adult male Cuviers beaked whale. Primary results reveal details of anatomic geometry with implications for acoustic function and diving. Specifically, we describe the juxtaposition of the large pterygoid sinuses, a fibrous venous plexus, and a lipid-rich pathway that connects the acoustic environment to the bony ear complex. We surmise that the large pterygoid air sinuses are essential adaptations for maintaining acoustic isolation and auditory acuity of the ears at depth. In the adult male, an acoustic waveguide lined with pachyosteosclerotic bones is apparently part of a novel transmission pathway for outgoing biosonar signals. Substitution of dense tissue boundaries where we normally find air sacs in delphinoids appears to be a recurring theme in deep-diving beaked whales and sperm whales. The anatomic configuration of the adult male Ziphius forehead resembles an upside-down sperm whale nose and may be its functional equivalent, but the homologous relationships between forehead structures are equivocal.

Quantity and distribution of levator ani stretch during simulated vaginal childbirth.

OBJECTIVE The objective of the study was to develop a model of the female pelvic floor to study levator stretch during simulated childbirth. STUDY DESIGN Magnetic resonance data from an asymptomatic nulligravida were segmented into pelvic muscles and bones to create a simulation model. Stiffness estimates of lateral and anteroposterior levator attachments were varied to estimate the impact on levator stretch. A 9 cm sphere was passed through the pelvis, along the path of the vagina, simulating childbirth. Levator response was interpreted at 4 positions of the sphere, simulating fetal head descent. The levator was color mapped to display the stretch experienced. RESULTS A maximum stretch ratio of 3.5 to 1 was seen in the posteriomedial puborectalis. Maximum stretch increased with increasing stiffness of lateral levator attachments. CONCLUSION Although preliminary, this work may help explain epidemiologic data regarding the pelvic floor impact of a first delivery. The models and simulation technique need refinement, but they may help study the effect of labor parameters on the pelvic floor.