Three time integration methods for incompressible flows with discontinuous Galerkin Boltzmann method

This paper presents three time integration methods for incompressible flows with finite element method in solving the lattice-BGK Boltzmann equation. The space discretization is performed using nodal discontinuous Galerkin method, which employs unstructured meshes with triangular elements and high order approximation degrees. The time discretization is performed using three different kinds of time integration methods, namely, direct, decoupling and splitting. From the storage cost, temporal accuracy, numerical stability and time consumption, we systematically compare three time integration methods. Then benchmark fluid flow simulations are performed to highlight efficient time integration methods. Numerical results are in good agreement with others or exact solutions.     

A triangular thin-shell finite element based on discrete Kirchhoff theory

A three-node, curved thin-shell triangular element with simple nodal connections is developed. The displacement and rotation components are independently interpolated by complete cubic and quadratic polynomials respectively. The Kirchhoff hypothesis is enforced at a discrete number of points in the element. The rigid-body displacement condition is satisfied by isoparametric interpolation of the shell geometry within the element. A detailed numerical evaluation through a number of standard problems is performed.     

Dynamic elastoplastic analysis of 3-D structures by the domain/boundary element method

The main objective of this paper is to present a general three-dimensional boundary element methodology for solving transient dynamic elastoplastic problems. The elastostatic fundamental solution is used in writing the integral representation and this creates in addition to the surface integrals, volume integrals due to inertia and inelasticity. Thus, an interior discretization in addition to the usual surface discretization is necessary. Isoparametric linear quadrilateral elements are used for the surface discretization and isoparametric linear hexahedra for the interior discretization. Advanced numerical integration techniques for singular and nearly singular integrals are employed. Houbolt's step-by-step numerical time integration algorithm is used to provide the dynamic response. Numerical examples are presented to illustrate the method and demonstrate its accuracy.     

Curved shell elements for heat conduction with p-approximation in the shell thickness direction

A finite element formulation is presented for the curved shell elements for heat conduction where the element temperature approximation in the shell thickness direction can be of an arbitrary polynomial order p. This is accomplished by introducing additional nodal variables in the element approximation corresponding to the complete Lagrange interpolating polynomials in the shell thickness direction. This family of elements has the important hierarchical property, i.e. the element properties corresponding to an approximation order p are a subset of the element properties corresponding to an approximation order p + 1. The formulation also enforces continuity or smoothness of temperature across the inter-element boundaries, i.e. C⁰ continuity is guaranteed.

The curved shell geometry is constructed using the co-ordinates of the nodes lying on the middle surface of the shell and the nodal point normals to the middle surface. The element temperature field is defined in terms of hierarchical element approximation functions, nodal temperatures and the derivatives of the nodal temperatures in the element thickness direction corresponding to the complete Lagrange interpolating polynomials. The weak formulation (or the quadratic functional) of the three-dimensional Fourier heat conduction equation is constructed in the Cartesian co-ordinate space. The element properties of the curved shell elements are then derived using the weak formulation (or the quadratic functional) and the hierarchical element approximation. The element matrices and the equivalent heat vectors (resulting from distributed heat flux, convective boundaries and internal heat generation) are all of hierarchical nature. The element formulation permits any desired order of temperature distribution through the shell thickness.

A number of numerical examples are presented to demonstrate the superiority, efficiency and accuracy of the present formulation and the results are also compared with the analytical solutions. For the first three examples, the h-approximation results are also presented for comparison purposes.     

Nonlinear finite element analysis of shells: Part I. three-dimensional shells

A nonlinear finite element formulation is presented for the three-dimensional quasistatic analysis of shells which accounts for large strain and rotation effects, and accommodates a fairly general class of nonlinear, finite-deformation constitutive equations. Several features of the developments are noteworthy, namely: the extension of the selective integration procedure to the general nonlinear case which, in particular, facilitates the development of a ‘heterosis-type’ nonlinear shell element; the presentation of a nonlinear constitutive algorithm which is ‘incrementally objective’ for large rotation increments, and maintains the zero normal-stress condition in the rotating stress coordinate system; and a simple treatment of finite-rotational nodal degrees-of-freedom which precludes the appearance of zero-energy in-plane rotational modes. Numerical results indicate the good behavior of the elements studied.     

Analysis and optimal design of plates and shells under dynamic loads – I: finite element and sensitivity analysis

In this paper we consider the development, integration, and application of reliable and efficient computational tools for the geometry modeling, mesh generation, structural analysis, and sensitivity analysis of variable-thickness plates and free-form shells under dynamic loads. A flexible shape-definition tool for surface modeling using Coons patches is considered to represent the shape and the thickness distribution of the structure, followed by an automatic mesh generator for structured meshes on the shell surface. Nine-node quadrilateral Mindlin–Reissner shell elements degenerated from 3D elements and with an assumed strain field, the so-called Huang–Hinton elements, are used for the FE discretization of the structure. The Newmark direct integration algorithm is used for the time discretization of the dynamic equilibrium equations for both the structural analysis and the semi-analytical (SA) sensitivity analysis. Alternatively, the sensitivities are computed by using the global finite difference (FD) method. Several examples are considered. In a companion paper, the tools presented here are combined with mathematical programming algorithms to form a robust and reliable structural optimization process to achieve better dynamic performance on the shell designs.     

Elastic torsion of thin walled bars of variable cross sections

An application of the finite element method to the theory of thin walled bars of variable cross sections has been presented in this paper. A solution of this problem is based on the linear membrane shell theory with the application of Vlasov's assumptions. A bar is divided into elements along its longitudinal axis and then, a shell mid-surface of the element is approximated by arbitrary triangular Subelements. Nodal displacements of the element are assumed to be polynomials of the third order and the equivalent stiffness matrix is obtained. Calculated nodal displacements enable an analysis of normal and shearing stresses.     

On a physically stabilized one point finite element formulation for three-dimensional finite elasto-plasticity

In the case of linear elasticity, a direct connection between the concept of reduced integration with hourglass stabilization and a mixed method can usually be established. In the non-linear case, this is in general not possible. To overcome this difficulty we suggest in this paper a new concept based on a Taylor expansion of the constitutively dependent quantities with respect to the centre of the element. The push-forward of the second (linear) term of the Taylor series for the first Piola–Kirchhoff stress tensor to the current configuration determines the so-called hourglass stabilization part of the residual force vector. Due to the fact that the element uses only one Gauss point and the hourglass stabilization part is computed by means of a simple functional evaluation, the present element technology is very efficient from the computational point of view.In contrast to the 2D case the computation of the Jacobi determinant only in the centre of the 3D element does not yield the correct volume, if the element shape deviates from being a parallelipiped. It is shown in the paper that the error becomes negligibly small for a relatively coarse discretization. The formulation is free of volumetric locking and can compete with shell formulations up to an aspect ratio of about hundred. For bending-dominated problems, at least two elements over the thickness are needed in order to compute the onset of plastification correctly. The element behaves very robustly in finite elasticity and inelasticity, also when large element distortions occur.     

Materially and geometrically nonlinear analysis of laminated anisotropic plates by p-version of FEM

A p-version finite element model based on degenerate shell element is proposed for the analysis of orthotropic laminated plates. In the nonlinear formulation of the model, the total Lagrangian formulation is adopted with moderately large deflections and small rotations being accounted for in the sense of von Karman hypothesis. The material model is based on the Huber-Mises yield criterion and Prandtl-Reuss flow rule in accordance with the theory of strain hardening yield function, which is generalized for anisotropic materials by introducing the parameters of anisotropy. The model is also based on the equivalent-single layer laminate theory. The integrals of Legendre polynomials are used for shape functions with p-level varying from 1 to 10. Gauss-Lobatto numerical quadrature is used to calculate the stresses at the nodal points instead of Gauss points. The validity of the proposed p-version finite element model is demonstrated through several comparative points of view in terms of ultimate load, convergence characteristics, nonlinear effect, and shape of plastic zone.     

A co-rotational formulation for nonlinear dynamic analysis of curved euler beam

A co-rotational finite element formulation for the dynamic analysis of a planar curved Euler beam is presented. The Euler-Bernoulli hypothesis and the initial curvature are properly considered for the kinematics of a curved beam. Both the deformational nodal forces and the inertial nodal forces of the beam element are systematically derived by consistent linearization of the fully geometrically nonlinear beam theory in element coordinates which are constructed at the current configuration of the corresponding beam element. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed here for the solution of the nonlinear dynamic equilibrium equations. Numerical examples are presented to demonstrate the effectiveness of the proposed element and to investigate the effect of the initial curvature on the dynamic response of the curved beam structures.     

Dynamic response of layered conical shell panel using integral equation technique

In this paper, the application of integral equation technique to dynamic response problem has been illustrated by considering the layered conical shell panel. The method consists of using the integral equation technique in the space domain and direct integration using the Wilson-theta method in the time domain. Since results are not available for layered conical shell panels, the values obtained for the particular case of isotropic cylindrical panel have been compared with the results obtained using a different procedure, viz series solution combined with mode superposition.     

Nonlinear elastoplastic dynamic analysis of single-layer reticulated shells subjected to earthquake excitation

A nonlinear dynamic finite element technique is developed to analyze the elastoplastic dynamic response of single-layer reticulated shells under strong earthquake excitation, in which the nonlinear three-dimensional beam elements are employed. An elastoplastic tangent stiffness matrix of three-dimensional beam element is derived by using the updated Lagrangian formulation, in which the isotropic hardening model, the Von-Mises yield criterion and the Prandtl-Reuss flow relations are applied to this study. This procedure considers both geometric and material nonlinearities. In this paper, several condensation and reduction techniques in matrices and degrees of freedom are used to simplify the analysis. An incremental-iterative technique based on the Newmark direct integration method and the modified Newton-Raphson method is employed for obtaining the solutions of the nonlinear dynamic equilibrium equations. Moreover, an accurate method is developed to compute the large rotations of space structures. As a numerical example, the elastoplastic dynamic response of a single-layer reticulated shell under strong seismic excitation is investigated. It is shown through the numerical example that the method developed in this paper is efficient for the nonlinear dynamic response analysis and plastic design of space structures.     

A discontinuous Galerkin method for the wave equation

We present a new discontinuous Galerkin method for solving the second-order wave equation using the standard continuous finite element method in space and a discontinuous method in time directly applied to second-order ode systems. We prove several optimal a priori error estimates in space–time norms for this new method and show that it can be more efficient than existing methods. We also write the leading term of the local discretization error in terms of Lobatto polynomials in space and Jacobi polynomials in time which leads to superconvergence points on each space–time cell. We discuss how to apply our results to construct efficient and asymptotically exact a posteriori estimates for space–time discretization errors. Numerical results are in agreement with theory.     

Large displacement extension of consistent shell element for static and dynamic analysis

The superior performance of the consistent shell element in the small deflection range has encouraged the authors to extend the formulation to large displacement static and dynamic analyses. The nonlinear extension is based on a total Lagrangian approach. A detailed derivation of the non-linear extension is based on a total Lagrangian approach. A detailed derivation of the non-linear stiffness matrix and the unbalanced load vector for the consistent shell element is presented in this study. Meanwhile, a simplified method for coding the nonlinear formulation is provided by relating the components for the nonlinear B-matrices to those of the linear B-matrix. The consistent mass matrix for the shell element is also derived and then incorporated with the stiffness matrix to perform large displacement dynamic and free vibration analyses of shell structures. Newmark's method is used for time integration and the Newton-Raphson method is employed for iterating within each increment until equilibrium is achieved. Numerical testing of the nonlinear model through static and dynamic analyses of different plate and shell problems indicates excellent performance of the consistent shell element in the nonlinear range.     

BDDC preconditioning for high-order Galerkin Least-Squares methods using inexact solvers

A high-order Galerkin Least-Squares (GLS) finite element discretization is combined with a Balancing Domain Decomposition by Constraints (BDDC) preconditioner and inexact local solvers to provide an efficient solution technique for large-scale, convection-dominated problems. The algorithm is applied to the linear system arising from the discretization of the two-dimensional advection–diffusion equation and Euler equations for compressible, inviscid flow. A Robin–Robin interface condition is extended to the Euler equations using entropy-symmetrized variables. The BDDC method maintains scalability for the high-order discretization of the diffusion-dominated flows, and achieves low iteration count in the advection-dominated regime. The BDDC method based on inexact local solvers with incomplete factorization and p = 1 coarse correction maintains the performance of the exact counterpart for the wide range of the Peclet numbers considered while at significantly reduced memory and computational costs.     

Transition finite elements with temperature and temperature gradients as primary variables for axisymmetric heat conduction

This paper presents finite element formulation for a special class of elements referred to as “transition finite elements” for axisymmetric heat conduction. The transition elements are necessary in applications requiring the use of both axisymmetric solid elements and axisymmetric shell elements. The elements permit transition from the solid portion of the structure to the shell portion of the structure. A novel feature of the formulation presented here is that nodal temperatures as well as nodal temperature gradients are retained as primary variables. The weak formulation of the Fourier heat conduction equation is constructed in the cylindrical co-ordinate system (r, z). The element geometry is defined in terms of the co-ordinates of the nodes as well as the nodal point normals for the nodes lying on the middle surface of the element. The element temperature field is approximated in terms of element approximation functions, nodal temperatures and the nodal temperature gradients. The properties of the transition elements are then derived using the weak formulation and the element temperature approximation. The formulation presented here permits linear temperature distribution through the element thickness. Convective boundaries as well as distributed heat flux is permitted on all four faces of the element. Furthermore, the element formulation also permits distributed heat flux and orthotropic material behaviour. Numerical examples are presented, first to illustrate the accuracy of the formulation and second to demonstrate its usefulness in practical applications. Numerical results are also compared with the theoretical solutions.     

Advances in the formulation of the rotation-free basic shell triangle

A family of rotation-free three node triangular shell elements is presented. The simplest element of the family is based on an assumed constant curvature field expressed in terms of the nodal deflections of a patch of four elements and a constant membrane field computed from the standard linear interpolation of the displacements within each triangle. An enhanced version of the element is obtained by using a quadratic interpolation of the geometry in terms of the six patch nodes. This allows to compute an assumed linear membrane strain field which improves the in-plane behaviour of the original element. A simple and economic version of the element using a single integration point is presented. The efficiency of the different rotation-free shell triangles is demonstrated in many examples of application including linear and non-linear analysis of shells under static and dynamic loads, the inflation and de-inflation of membranes and a sheet stamping problem.     

Implicit upper bound error estimates for combined expansive model and discretization adaptivity

A computational methodology for goal-oriented combined discretization and expansive (refined) model adaptivity by overall implicit error control of quantities of interest is presented, requiring estimators of primal and dual discretization and model errors. In the case of dimensional within model adaptivity, prolongations of coarse model solutions into the solution space of a fine model for defining a consistent model error are necessary, which can be achieved at the element level by two strategies. The first one is an orthogonalized kinematic prolongation of nodal displacements, whereas the second one uses prolongations of the external loads which are then used to solve additional local variational problems thus yielding prolongated solutions which a priori fulfill the required orthogonality relations at the element level. Finally, a numerical example of an elastic continuous T-beam is presented with comparative results where goal-oriented error estimation is applied to linear elasticity with a $2\frac{1}{2}\text{D}$ discrete Reissner–Mindlin plate model as the coarse model and the 3D theory as the fine model.     

A large deformation,rotation-free,isogeometric shell

Conventional finite shell element formulations use rotational degrees of freedom to describe the motion of the fiber in the Reissner–Mindlin shear deformable shell theory, resulting in an element with five or six degrees of freedom per node. These additional degrees of freedom are frequently the source of convergence difficulties in implicit structural analyses, and, unless the rotational inertias are scaled, control the time step size in explicit analyses. Structural formulations that are based on only the translational degrees of freedom are therefore attractive. Although rotation-free formulations using C⁰ basis functions are possible, they are complicated in comparison to their C¹ counterparts. A C^k-continuous, k ? 1, NURBS-based isogeometric shell for large deformations formulated without rotational degrees of freedom is presented here. The effect of different choices for defining the shell normal vector is demonstrated using a simple eigenvalue problem, and a simple lifting operator is shown to provide the most accurate solution. Higher order elements are commonly regarded as inefficient for large deformation analyses, but a traditional shell benchmark problem demonstrates the contrary for isogeometric analysis. The rapid convergence of the quadratic element is demonstrated for the NUMISHEET S-rail benchmark metal stamping problem.     

Robustness of a Spline Element Method with Constraints

The spline element method with constraints is a discretization method where the unknowns are expanded as polynomials on each element and Lagrange multipliers are used to enforce the interelement conditions, the boundary conditions and the constraints in numerical solution of partial differential equations. Spaces of piecewise polynomials with global smoothness conditions are known as multivariate splines and have been extensively studied using the Bernstein-Bézier representation of polynomials. It is used here to write the constraints mentioned above as linear equations. In this paper, we illustrate the robustness of this approach on two singular perturbation problems, a fourth order problem and a Stokes-Darcy flow. It is shown that the method converges uniformly in the perturbation parameter.