P-type Finite Element Research Articles

LES/DNS fluid-structure interaction simulation of non-linear slender structures in Nektar++ framework

Nektar++ is a spectral/hp element open-source framework written in C++ for the construction of classical low-order h-type as well as higher-order p-type finite element solvers. It seeks to overcome the implementation challenges of the complex data structures associated with high-order finite element methods; hence, providing an efficient, flexible and HPC scalable platform for the development of solvers for partial differential equations using the spectral/hp element method. In the present work, capabilities of Nektar++ is leveraged for development of two fluid-structure interaction (FSI) solvers for simulations of highly deformable nonlinear slender structures. The FSI solver uses the incompressible Navier-Stokes (NS) solvers of Nektar++ for fluid flow while the structural dynamics is modelled using Geometrically-Exact Composite Beams (GECB). The open-source SHARPy framework is linked to Nektar++ and used for structural simulation. Aiming at high-fidelity (LES/DNS) FSI simulations, the thick-strip approach is used to reduce computational costs. In this approach, the full 3D fluid domain is represented with series of smaller 3D domains normal to the local axis of the structure and having a finite thickness in the spanwise direction where periodicity is also assumed. Hence, while reducing the computational costs by avoiding the discretization of equations over the entire slender structure, the strip thickness allows capturing the local 3D turbulent wake and accurately predict the fluid forces on the structure. Two approaches are adopted to avoid the dynamic remeshing due to the large and non-linear deformation of the structure. In the first approach, the transformed the NS equations are solved in the non-inertial body-fitted coordinates while in the second approach the NS equations are formulated in the moving frame of reference and solved with the spectral/hp element method. A hybrid parallelisation approach of Nektar++ is extended for the thick-strip method which allows having non-constant cross-section along the structural span as well as efficient and flexible use of computational resources, and excellent HPC performance for the FSI simulations. The capability of the FSI solver is demonstrated via several examples.

Multiscale Modeling of O2 Transport in Articular Cartilage

Partial pressure of oxygen (pO2) is suggested to have a regulatory effect on chondrocyte biosynthetic activities and its effect during expansion is unknown. We hypothesize that oxygen tension due to mechanical deformation or swelling could be as important as direct mechanical effects on cell biosynthetic activities. While there are plenty of studies on measuring and/or modeling pO2 in articular cartilage (AC) for static (rest) conditions, to our best of knowledge, there are no such studies on pO2 in AC for dynamic conditions such as swelling or tissue deformation. In this study, we attempt to develop a model to study the dynamics of oxygen transport in AC. We design a high-precision hybrid element using the p-type finite element method by which both, diffusion and convection, are incorporated as a single element. We use a domain decomposition method that allow us to use a different type of discretization with independent discretization variables in non-overlapping sub-domains, for a generic three-dimensional approach to elliptic boundary value problems of order two or higher. Results strongly support the idea that during normal activity, the partial pressure of oxygen in AC is uniform and high (close to 50 mmHg) unlike in static conditions where partial pressure has a large gradient from surface to bottom. The formulation developed in this study might be used in determining the necessary flow conditions to cultivate tissue constructs in tissue repair and tissue engineering.

A p-type finite element solution for the simulation of O2 transport in articular cartilage tissue: heterogeneous and porous media

The partial pressure of oxygen (pO2) is suggested to have a regulatory effect on chondrocyte biosynthetic activities, and its effect during expansion is unknown. The authors hypothesize that oxygen tension due to mechanical deformation or swelling could be as important as direct mechanical effects on cell biosynthetic activities. While there are plenty of studies on measuring and/or modelling pO2 in articular cartilage (AC) for static (rest) conditions, to the best of the authors' knowledge there are very few such studies on pO2 in AC for dynamic conditions such as swelling or tissue deformation. In this study, it is attempted to develop a model to study the dynamics of oxygen transport in AC. A high-precision hybrid element is designed using the p-type finite element method, by which diffusion and convection are incorporated as a single element. A domain decomposition method is used that allows the use of a different type of discretization with independent discretization variables in non-overlapping sub-domains, for a generic three-dimensional approach to elliptic boundary value problems of order 2 or higher. The formulation developed in this study might be used in determining the necessary flow conditions to cultivate tissue constructs in tissue repair and tissue engineering.

Advanced modeling strategy for the analysis of heart valve leaflet tissue mechanics using high-order finite element method

Modeling soft tissue using the finite element method is one of the most challenging areas in the field of biomechanical engineering. To date, many models have been developed to describe heart valve leaflet tissue mechanics, which are accurate to some extent. Nevertheless, there is no comprehensive method to modeling soft tissue mechanics, This is because (1) the degree of anisotropy in the heart valve leaflet changes layer by layer due to a variety of collagen fiber densities and orientations that cannot be taken into account in the model and also (2) a constitutive material model fully describing the mechanical properties of the leaflet structure is not available in the literature. In this framework, we develop a new high-order element using p-type finite element formulation to create anisotropic material properties similar to those of the heart valve leaflet tissue in only one single element. This element also takes the nonlinearity of the leaflet tissue into consideration using a bilinear material model. This new element is composed a two-dimensional finite element in the principal directions of leaflet tissue and a p-type finite element in the direction of thickness. The proposed element is easy to implement, much more efficient than standard elements available in commercial finite element packages. This study is one step towards the modeling of soft tissue mechanics using a meshless finite element approach to be applied in real-time haptic feedback of soft-tissue models in virtual reality simulation.

Fast Schwarz-type finite element electrostatic solver

A new p-type finite element solver is presented for the efficient solution of 3D electrostatic problems. Its efficiency comes from a careful implementation of multigrid ideas within the finite element framework. A multigrid-type preconditioner is developed for the Conjugate Gradient method, which leads to the overall O(N/sup 1.1/) complexity.

Least-squares p-r finite element methods for incompressible non-Newtonian flows

A least-squares mixed p-type finite element method is developed for numerical solution of incompressible non-Newtonian flows. Hierarchical piecewise polynomials are introduced as element basis functions while singularities and boundary layers are treated by a combination of mesh redistribution and polynomial refinement. Scaling of the original differential equations is found to be important for the least-squares minimization process. We discuss both nonlinear algebraic and differential constitutive models and present numerical examples to illustrate benefits and shortcomings of the present approach.

Stream function-vorticity driven cavity solution using p finite elements

Calculations for the two-dimensional driven cavity incompressible flow problem are presented. A p-type finite element scheme for the fully coupled stream function-vorticity formulation of the Navier-Stokes equations is used. Graded meshes are used to resolve vortex flow features and minimize the impact of corner singularities. Incremental continuation in the Reynolds number allows solutions to be computed for Re = 12 500. A significant feature of the work is that new tertiary and quaternary corner vortex features are observed in the flow field. Comparisons are made with other solutions in the literature.

COMPARISON OFH ANDP FINITE ELEMENT APPROXIMATIONS OF THE SHALLOW WATER EQUATIONS

SUMMARY A p-type finite element scheme is introduced for the three-dimensional shallow water equations with a harmonic expansion in time. The wave continuity equation formulation is used which decouples the problem into a Helmholtz equation for surface elevation and a momentum equation for horizontal velocity. An exploration of the applicability of p methods to this form of the shallow water problem is presented, with a consideration of the problem of continuity errors. The convergence rates and relative computational efficiency between h- and p-type methods are compared with the use of three test cases representing various degrees of difficulty. A channel test case establishes convergence rates, a continental shelf test case examines a problem with accuracy difficulties at the shelf break, and a field-scale test case examines problems with highly irregular grids. For the irregular grids, adaptive h combined with uniform p refinement was necessary to retain high convergence rates.

A triangular spectral element method; applications to the incompressible Navier-Stokes equations

Encouraged by the success of spectral elements methods in computational fluid dynamics and p-type finite element methods in structural mechanics we wish to extend these ideas to solving high order polynomial approximations on triangular domains as the next generation of spectral element solvers. We introduce here a complete formulation using a modal basis which has been implemented in a new codeNεκTαr. The new basis has the following properties: Jacobi polynomials of mixed weights; semi-orthogonality; hierarchical structure; generalized tensor (warped) product; variable order; and a new apex co-ordinate system allowing automated integration with Gaussian quadrature. We have discussed the formulation using a matrix notation which allows for an easy interpretation of the forward and backward transformations. We use this notation to formulate the linear advection and Helmholtz equations in an efficient manner and show that we recover the following properties: well conditioned matrices; asymptotic operation count of O(N3); scaling O(N2 of the spectral radius of the weak convective operator; and exponential convergence using polynomials up to ∼ 40. Having constructed and numerically analysed these equations we are then able to solve the incompressible Navier-Stokes equations using a high-order splitting scheme. We demonstrate a variety of results showing exponential convergence using both deformed and straight triangular subdomains for both the Stokes and Navier-Stokes problems.

Modal Analysis of a Suspension Assembly

The dynamic characteristics of a suspension assembly are examined using new numerical and experimental techniques. The p-type finite element method is used to construct a numerical model of the suspension. There are significant advantages in using this approach to analyze these types of structures. The model is verified by an experimental modal analysis system, which has been shown to be effective in the study of small structures. The modelled modal parameters agree within 4.5 percent with the experimental results for 14 modes. Since the experimental system uses an electromagnetic exciter, a ferromagnetic target must be attached to the nonferrous suspension so that it can be excited. Innovative techniques are investigated to improve the attachment of this ferromagnetic target. Furthermore, the finite element model is utilized to evaluate the sensitivity of the modal parameters of the suspension to changes in its geometrical features.

Tuned Constrained Layer Damping Of A Cantilevered Plate

A damping patch is shown to behave as a constrained layer damper (CLD) for the bending modes of a tapered cantilevered plate, while behaving as a tuned damper for the sway modes. This dual nature leads to higher damping capacity for the sway modes than would be otherwise possible using a similar coverage of conventional CLD, at the expense of optimum CLD performance in bending. An added variable end mass allows determination of the frequency dependence of experimentally measured loss factors and those calculated using a finite element model. Measured loss factor frequency dependence is compared with predictions from CLD theory and tuned damper theory. The plate response is measured using laser Doppler vibrometry for out-of-plane motion, and laser Doppler anemometry for in-plane motion, while force excitation is measured with a piezoelectric transducer. Modal analysis of measured mobility frequency response functions yields experimentally determined mode shapes, frequencies, and loss factors. A p-type finite element code, in conjunction with the modal strain energy method, is used to perform a normal modes and damping analysis of the damped plate geometry for comparison with experimental results.

Viscoelastic Damping Calculations Using a p-Type Finite Element Code

Damping factors of viscoelastically damped structures can be calculated using the modal strain energy method, implemented with a sequence of undamped modal analysis computations. There are significant advantages in performing these calculations using p-type finite element codes. These include ease of mesh design, an indicator of degree of solution convergence, modest computation time, and an insensitivity to element aspect ratio. Capitalizing on these advantages an algorithm is defined, which is effective in solving for the natural frequencies and modal loss factors of damped structures. The algorithm is demonstrated using a sandwiched cantilevered beam as an example.

Adaptive solution strategy for solving large systems ofp-type finite element equations

A solution strategy is proposed and implemented for taking advantage of the hierarchical structure of linear equation sets arising from the p-type finite element method using a hierarchical basis function set. The algorithm dynamically branches to either direct or iterative solution methods. In. the iterative solution branch, the substructure of the finite element equation set is used to generate a lower order preconditioner for a preconditioned conjugate gradient (PCG) method. The convergence rate of the PCG algorithm is monitored to improve the heuristics used in the choice of the preconditioner. The robustness and efficiency of the method are demonstrated on a variety of three dimensional examples utilizing both hexahedral and tetrahedral mesh discretizations. This strategy has been implemented in a p-version finite element code which has been used in an industrial environment for over two years to solve mechanical design problems.

Singularities in p-Type Finite Element Computations: Numerical Experiments on the Locality of Pollution Effects

Using a recently developed code for three-dimensional p-type finite element computations in elasticity, we have studied, with a series of numerical experiments, the behavior of the solution in the presence of singularities. Worst-cast theoretical estimates predict a global pollution of the solution, unless optimally graded meshes are used. On the other hand, we observe that except in an immediate neighborhood of a singularity, and even using uniform meshes, the quality of the solution is not degraded to any practically relevant extent. This has important practical consequences because it allows modelers to introduce artifical singularities for the purpose of simplifying models. The size of the area around a singularity where the solution is appreciably degraded can be estimated in terms of the minimum effective wavelength of the basis functions used; the latter, in turn, can be related to the element size and the polynomial order used.