Postprocessing techniques and h-adaptive finite element-eigenproblem analysis

Hydrated soft tissues of the human musculoskeletal system, such as articular cartilage in diarthrodial joints or the annulus fibrosis of the inter vertebral disk, are accurately represented by a biphasic continuum model consisting of an incompressible solid phase (collagen and proteoglycan) and incompressible, inviscid fluid (interstitial fluid) and derive from the continuum theory of mixtures. These tissues exhibit a viscoelastic‐type response which is caused by the diffusive drag of the fluid phase as it flows past the solid phase. In this study an automated, adaptive, finite element solution of the governing biphasic equations is presented. The finite element formulation is based on a mixed‐penalty approach in which the penalty form of the continuity equation for the mixture is included in the weak form. Pressure, solid and fluid velocities are interpolated independently, and the coefficients of the C−1 pressure field are eliminated at the element level. The resulting matrix form is a system of first order differential equations which is solved via standard finite difference methods. Mesh generation and updating, including both refinement and coarsening, for the two‐dimensional examples examined in this study are performed using Finite Quadtree. The adaptive analysis is based on an error indicator which is the L2 norm of the difference between the finite element solution and a projected finite element solution. Total stress, calculated as the sum of the solid and fluid phase stresses, is used in the error indicator. Rezoning is accomplished by transferring the finite element solution for the primary variables onto the locally updated mesh using a projected field. These projected values allow the finite difference algorithm to proceed in time using the updated mesh. The accuracy and effectiveness of this adaptive finite element analysis is demonstrated using two‐dimensional axisymmetric problems corresponding to the uncomined compression of a cylindrical disk of soft tissue and the indentation of a thin sheet of soft tissue. The method is shown to effectively capture the steep gradients in both problems and to produce solutions in good agreement with independent, converged, numerical solutions.

We consider the construction of locally conservative fluxes by means of a simple postprocessing technique obtained from the finite element solutions of advection diffusion equations. It is known that a naive calculation of fluxes from these solutions yields nonconservative fluxes. We consider two finite element methods: the usual continuous Galerkin finite element method for solving nondominating advection diffusion equations and the streamline upwind/Petrov‐Galerkin method for solving advection dominated problems. We then describe the postprocessing technique for constructing conservative fluxes from the numerical solutions of the general variational formulation. The postprocessing technique requires solving an auxiliary Neumann boundary value problem on each element independently and it produces a locally conservative flux on a vertex centered dual mesh relative to the finite element mesh. We provide a convergence analysis for the postprocessing technique. Performance of the technique and the convergence behavior are demonstrated through numerical examples including a set of test problems for advection diffusion equations, advection dominated equations, and drift‐diffusion equations. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1971–1994, 2015

A spectral-Tchebychev solution for three-dimensional dynamics of curved beams under mixed boundary conditions

This paper presents a new adaptive finite element (FE) procedure for accurate, efficient, and reliable computation of the critical buckling loads and the associated modes of nonuniform Bernoulli-Euler members. After the conventional FE solution on a given mesh has been obtained, a novel conceptual and practical strategy is introduced, in which the FE solution of the eigenproblem is equivalently viewed as the FE solution of an associated linear problem. This strategy allows the recently developed element energy projection (EEP) technique to be readily used to calculate super-convergent FE solutions for buckling modes, which are subsequently used to estimate the FE solution errors and to further guide mesh refinements, until the accuracy of the FE solution matches the user-preset error tolerance. Numerical examples are given to show the effectiveness, efficiency, accuracy, and reliability of the proposed method, which also paves the way for development of an adaptive FE solver for more general and complicated eigenproblems.

Improved formulation for spatial stability and free vibration of thin-walled tapered beams and space frames

A split-step finite-element method for incompressible Navier-Stokes equations with high-order accuracy up-to the boundary

In order to assess the discretization error of a finite element solution, asymptotic solutions for predicted natural frequencies of two-dimensional elastic solid vibration problems in the finite element analysis are presented in this paper. Since the asymptotic solution is more accurate than the original finite element solution, it can be viewed as an alternative solution against which the original finite element solution can be compared. Consequently, the discretization error of the finite element solution can be evaluated. Due to the existence of two kinds of two-dimensional problems in engineering practice, both the plane stress problem and the plane strain problem have been considered and the corresponding asymptotic formulae for predicted natural frequencies of two-dimensional solids by the finite element method have been derived from the fact that a discretized finite element system approaches a continuous one if the finite element size approaches zero. It has been demonstrated, from the related numerical results of three examples, that the present asymptotic solution, which can be obtained by simply using the corresponding formula without any further finite element calculation, is indeed more accurate than the original finite element solution so that it can be considered as a kind of corrected solution for the discretization error estimation of a finite element solution.

A comprehensive finite element model incorporating the features of comprehensive wing model and finite element method is developed in this paper. This new model extends the applicability of comprehensive wing model from uniform composite wings to tapered composite wings. It preserves the simplicity of one-dimensional finite element but performs like three-dimensional finite element for wing structural analysis. To verify its accuracy and show its efficiency, this model which is applicable to the general dynamic analysis is reduced to two simple cases: static analysis and free vibration analysis. Based upon the reduction results for static and free vibration cases, several numerical examples of composite wing structures are presented such as tapered wings subjected to end load or uniform pressure, and free vibration of tapered wings. The numerical results show that the proposed model is much more efficient than the conventional finite element, and their differences are within 12% in all cases of composite wing structures.

A comparison of the non-conforming and conforming sector finite element for free vibration of circular discs

A hyperbolic equation is considered for the propagation of pressure disturbance waves in layered fluids having different fluid properties. For acoustic problems of this sort, the characteristic finite element model alone does not suffice to ensure prediction of the monotonic wave profile across fluids having different properties. A flux corrected transport solution algorithm is intended for incorporation into the underlying Taylor–Galerkin finite element framework. The advantage of this finite element approach, in addition to permitting oscillation-free solutions, is that it avoids the necessity of dealing with medium discontinuity. As an analysis tool, the proposed monotonic finite element model has been intensively verified through problems which are amenable to analytic solutions. In modeling wave propagation in layered fluids, we have investigated the influence of the degree of medium change on the finite element solutions. Also, different finite element solutions are considered to show the superiority of using the flux corrected transport Taylor–Galerkin finite element model.

High precision magnetic field computer programs are required to accurately calculate the multipole content and magnetic field strength associated with a given accelerator magnet design. In this work, a specialized version of the finite element computer program WEMAP (Westinghouse Electric and Magnetic Analysis Program) is applied to field harmonic calculations. The computer program relies on a higher-order finite element formulation and high-precision computer representation to minimize calculation error. Several calculations are made for a SSC CDM cross section design. The effects of mesh fineness and finite element order are investigated by examining finite element grids with increasing mesh density and element order. A key feature of this computer program is the capability to minimize the truncation error associated with a finite element mesh by increasing the order of the finite element interpolating polynomial. Finite elements of order one through six are incorporated. Inaccuracies which may occur using the lower-order finite elements with a coarse mesh are identified in this paper. Care must be taken in meshing the yoke iron region to accurately account for field harmonics due to nonuniform iron permeability. Methods of extracting the spatial harmonics from the finite element solution are discussed. The field strength and harmonics are calculated for a full-section model of a CDM magnet cross section.

Given the functional complexities of soft tissues and organs, it is clear that computational simulations are critical in their understanding and for the rational basis for the development of therapies and replacements. A key aspect of such simulations is accounting for their complex, nonlinear, anisotropic mechanical behaviors. While soft tissue material models have developed to the point of high fidelity, in-silico implementation is typically done using the finite element (FE) method, which remains impractically slow for translational clinical time frames. As a potential path toward addressing the development of high fidelity simulations capable of performing in clinically relevant time frames, we review the use of neural networks (NN) for soft tissue and organ simulation using two approaches. In the first approach, we show how a NN can learn the responses for a detailed meso-structural soft tissue material model. The NN material model not only reproduced the full anisotropic mechanical responses but also demonstrated a considerable efficiency improvement, as it was trained over a range of realizable fibrous structures. In the second approach, we go a step further with the use of a physics-based surrogate model to directly learn the displacement field solution without the need for raw training data or FE simulation datasets. In this approach we utilize a finite element mesh to define the domain and perform the necessary integrations, but not the finite element method (FEM) itself. We demonstrate with this approach, termed neural network finite element (NNFE), results in a trained NNFE model with excellent agreement with the corresponding "ground truth" FE solutions over the entire physiological deformation range on a cuboidal myocardium specimen. More importantly, the NNFE approach provided a significantly decreased computational time for a range of finite element mesh sizes. Specifically, as the FE mesh size increased from 2744 to 175,615 elements, the NNFE computational time increased from 0.1108 s to 0.1393 s, while the "ground truth" FE model increased from 4.541 s to 719.9 s, with the same effective accuracy. These results suggest that NNFE run times are significantly reduced compared with the traditional large-deformation-based finite element solution methods. We then show how a nonuniform rational B-splines (NURBS)-based approach can be directly integrated into the NNFE approach as a means to handle real organ geometries. While these and related approaches are in their early stages, they offer a method to perform complex organ-level simulations in clinically relevant time frames without compromising accuracy.

One potential application of aluminium foam sandwich panels in civil engineering is the cladding system which is employed to protect other structures again impact and blast loadings. Finite element (FE) simulation of these sandwich panels subjected to impact loading was conducted by using a commercial software package, LS-DYNA (version 971). The FE model was verified by experimental results conducted previously. Good agreement was achieved between the FE and experimental results. Parametric study was conducted to investigate the effects of skin thickness, core thickness and boundary conditions on the deformation modes and energy absorption of sandwich panels with aluminum foam core.

An elasto-plastic damage model for functionally graded plates with in-plane material properties variation: Material model and numerical implementation

Dynamic stiffness method and CUF-based component-wise theories applied to free vibration analysis of solid beams, thin-walled structures and reinforced panels