Primal and Shadow functions, Dual and Dual-Shadow functions for a circular crack and a circular 90 degree V-notch with Neumann boundary conditions

This report presents explicit analytical expressions for the primal, primal shadows, dual and dual shadows functions for the Laplace equation in the vicinity of a circular singular edge with Neumann boundary conditions on the faces that intersect at the singular edge. Two configurations are investigated: a penny-shaped crack and a 90^o V-notch

A Parallel High-Order Fictitious Domain Approach for Biomechanical Applications

The focus of this contribution is on the parallelization of the Finite Cell Method (FCM) applied for biomechanical simulations of human femur bones. The FCM is a high-order fictitious domain method that combines the simplicity of Cartesian grids with the beneficial properties of hierarchical approximation bases of higher order for an increased accuracy and reliablility of the simulation model. A pre-computation scheme for the numerically expensive parts of the finite cell model is presented that shifts a significant part of the analysis update to a setup phase of the simulation, thus increasing the update rate of linear analyses with time-varying geometry properties to a range that even allows user interactive simulations of high quality. Paralellization of both parts, the pre-computation of the model stiffness and the update phase of the simulation is simplified due to a simple and undeformed cell structure of the computation domain. A shared memory parallelized implementation of the method is presented and its performance is tested for a biomedical application of clinical relevance to demonstrate the applicability of the presented method

High frequency oscillations of first eigenmodes in axisymmetric shells as the thickness tends to zero

The lowest eigenmode of thin axisymmetric shells is investigated for two physical models (acoustics and elasticity) as the shell thickness (2$\epsilon$) tends to zero. Using a novel asymptotic expansion we determine the behavior of the eigenvalue $\lambda$($\epsilon$) and the eigenvector angular frequency k($\epsilon$) for shells with Dirichlet boundary conditions along the lateral boundary, and natural boundary conditions on the other parts. First, the scalar Laplace operator for acoustics is addressed, for which k($\epsilon$) is always zero. In contrast to it, for the Lam{\'e} system of linear elasticity several different types of shells are defined, characterized by their geometry, for which k($\epsilon$) tends to infinity as $\epsilon$ tends to zero. For two families of shells: cylinders and elliptical barrels we explicitly provide $\lambda$($\epsilon$) and k($\epsilon$) and demonstrate by numerical examples the different behavior as $\epsilon$ tends to zero

Uncertainty quantification for personalized analyses of human proximal femurs

Computational models for the personalized analysis of human femurs contain uncertainties in bone material properties and loads, which affect the simulation results. To quantify the influence we developed a probabilistic framework based on polynomial chaos (PC) that propagates stochastic input variables through any computational model. We considered a stochastic E-ρ relationship and a stochastic hip contact force, representing realistic variability of experimental data. Their influence on the prediction of principal strains (ϵ1 and ϵ3) was quantified for one human proximal femur, including sensitivity and reliability analysis. Large variabilities in the principal strain predictions were found in the cortical shell of the femoral neck, with coefficients of variation of Math Eq. Between 60-80% of the variance in ϵ1 and ϵ3 are attributable to the uncertainty in the E-ρ relationship, while Math Eq are caused by the load magnitude and 5-30% by the load direction. Principal strain directions were unaffected by material and loading uncertainties. The antero-superior and medial inferior sides of the neck exhibited the largest probabilities for tensile and compression failure, however all were very small (Math Eq). In summary, uncertainty quantification with PC has been demonstrated to efficiently and accurately describe the influence of very different stochastic inputs, which increases the credibility and explanatory power of personalized analyses of human proximal femurs

Free Vibrations of Axisymmetric Shells: Parabolic and Elliptic cases

Approximate eigenpairs (quasimodes) of axisymmetric thin elastic domains with laterally clamped boundary conditions (Lam{\'e} system) are determined by an asymptotic analysis as the thickness ($2\varepsilon$) tends to zero. The departing point is the Koiter shell model that we reduce by asymptotic analysis to a scalar modelthat depends on two parameters: the angular frequency $k$ and the half-thickness $\varepsilon$. Optimizing $k$ for each chosen $\varepsilon$, we find power laws for $k$ in function of $\varepsilon$ that provide the smallest eigenvalues of the scalar reductions.Corresponding eigenpairs generate quasimodes for the 3D Lam{\'e} system by means of several reconstruction operators, including boundary layer terms. Numerical experiments demonstrate that in many cases the constructed eigenpair corresponds to the first eigenpair of the Lam{\'e} system.Geometrical conditions are necessary to this approach: The Gaussian curvature has to be nonnegative and the azimuthal curvature has to dominate the meridian curvature in any point of the midsurface. In this case, the first eigenvector admits progressively larger oscillation in the angular variable as $\varepsilon$ tends to $0$. Its angular frequency exhibits a power law relationof the form $k=\gamma \varepsilon^{-\beta}$ with $\beta=\frac14$ in the parabolic case (cylinders and trimmed cones), and the various $\beta$s $\frac25$, $\frac37$, and $\frac13$ in the elliptic case.For these cases where the mathematical analysis is applicable, numerical examples that illustrate the theoretical results are presented

Phase-field boundary conditions for the voxel finite cell method: surface-free stress analysis of CT-based bone structures

The voxel finite cell method employs unfitted finite element meshes and voxel quadrature rules to seamlessly transfer CT data into patient-specific bone discretizations. The method, however, still requires the explicit parametrization of boundary surfaces to impose traction and displacement boundary conditions, which constitutes a potential roadblock to automation. We explore a phase-field based formulation for imposing traction and displacement constraints in a diffuse sense. Its essential component is a diffuse geometry model generated from metastable phase-field solutions of the Allen-Cahn problem that assumes the imaging data as initial condition. Phase-field approximations of the boundary and its gradient are then employed to transfer all boundary terms in the variational formulation into volumetric terms. We show that in the context of the voxel finite cell method, diffuse boundary conditions achieve the same accuracy as boundary conditions defined over explicit sharp surfaces, if the inherent length scales, i.e., the interface width of the phase-field, the voxel spacing and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human femur and a vertebral body

Extracting generalized edge flux intensity functions with the quasidual function method along circular 3-D edges

International audienceExplicit asymptotic series describing solutions to the Laplace equation in the vicinity of a circular edge in a three-dimensional domain was recently provided in Yosibash et al, Int. Jour. Fracture, 168 (2011), pp. 31-52. Utilizing it, we extend the quasidual function method (QDFM) for extracting the generalized edge flux intensity functions (GEFIFs) along circular singular edges in the cases of axisymmetric and non-axisymmetric data. This accurate and efficient method provides a functional approximation of the GEFIFs along the circular edge whose order is adaptively increased so to approximate the exact GEFIFs. It is implemented as a post-solution operation in conjunction with the p-version of the finite element method. The mathematical analysis of the QDFM is provided, followed by numerical investigations, demonstrating the efficiency, robustness and high accuracy of the proposed quasi-dual function method. The mathematical machinery developed in the framework of the Laplace operator is important to realize its possible extension for the elasticity system

The p-version of the finite element method in incremental elasto-plastic analysis

Whereas the higher-order versions of the finite elements method (the p- and hp-version) are fairly well established as highly efficient methods for monitoring and controlling the discretization error in linear problems, little has been done to exploit their benefits in elasto-plastic structural analysis. Aspects of incremental elasto-plastic finite element analysis which are particularly amenable to improvements by the p-version is discussed. These theoretical considerations are supported by several numerical experiments. First, an example for which an analytical solution is available is studied. It is demonstrated that the p-version performs very well even in cycles of elasto-plastic loading and unloading, not only as compared to the traditional h-version but also in respect to the exact solution. Finally, an example of considerable practical importance - the analysis of a cold-worked lug - is presented which demonstrates how the modeling tools offered by higher-order finite element techniques can contribute to an improved approximation of practical problems

Circular edge singularities for the Laplace equation and the elasticity system in 3-D domains

International audienceAsymptotics of solutions to the Laplace equation with Neumann or Dirichlet conditions in the vicinity of a circular singular edge in a three-dimensional domain are derived and provided in an explicit form. These asymptotic solutions are represented by a family of eigen-functions with their shadows, and the associated edge flux intensity functions (EFIFs), which are functions along the circular edge. We provide explicit formulas for a penny-shaped crack for an axisymmetric case as well as a case in which the loading is non-axisymmetric. Explicit formulas for other singular circular edges such as a circumferential crack and an external crack are also derived. The mathematical machinery developed in the framework of the Laplace operator is extended to derive the asymptotic solution (three-component displacement vector) for the elasticity system in the vicinity of a circular edge in a three-dimensional domain. As a particular case we present explicitly the series expansion for a traction free or clamped penny-shaped crack in an axisymmetric or a non-axisymmetric situation. The precise representation of the asymptotic series is required for constructing benchmark problems with analytical solutions against which numerical methods can be assessed, and to develop new extraction techniques for the edge flux/intensity functions which are of practical engineering importance in predicting crack propagation. <br

Edge Stress Intensity Functions in Polyhedral Domains and their Extraction by a Quasidual Function Method

International audienceThe solution to elastic isotropic problems in three-dimensional (3-D) polyhedral domains has an explicit structure in the vicinity of the edges. This structure involves a family of eigen-functions with their shadows, and the associated Edge Stress Intensity Functions (ESIFs), which are functions along the edges. For the extraction of ESIFs, we apply the Quasidual Function Method, the theoretical fundation of which is due to a former work by MC, MD and ZY. This method provides a polynomial approximation of the ESIF along the edge whose order is adaptively increased so to approximate the exact ESIF. It is implemented as a post-solution operation in conjunction with the p-version finite element method. Numerical examples are provided in which we extract ESIFs associated with traction free or homogeneous Dirichlet boundary conditions in 3-D cracked domains or 3-D V-Notched domains. These demonstrate the efficiency, robustness and high accuracy of the proposed quasi-dual function method