Combination of Hexahedral and Prismatic Solid-shell Finite Elements

In this study, we developed a neutron diffusion equation solver based on the finite element method for CAPP code. Three types of triangular finite elements and five types of rectangular depending on the order of the shape functions were implemented for 2-D application. Ten types of triangular prismatic finite elements and seventeen types of rectangular prismatic finite elements were also implemented for 3-D application. Two types of polynomial mapping from the master finite element to a real finite element were adopted for flexibility in dealing with complex geometry. They are linear mapping and iso-parametric mapping. In linear mapping, only the vertex nodes are used as the mapping points. In iso-parametric mapping, all the nodal points in the finite element are used as the mapping points, which enables the real finite elements to have curved surfaces. For the treatment of spatial dependency of cross-sections in the finite elements, three types of polynomial expansion of the cross-sections in the finite elements were implemented. They are constant, linear, and iso-parametric cross-section expansions. The power method with the Wielandt acceleration technique was adopted as the outer iteration algorithm. The BiCGSTAB algorithm with the ILU (Incomplete LU) decomposition preconditioner was used as the linear equation solver in the inner iteration. The neutron diffusion equation solver developed in this study was verified against two well known benchmark problems, IAEA PWR benchmark problem and OECD/NEA PBMR400 benchmark problem. Results of numerical tests showed that the solution converged to the reference solution as the finite elements are refined and as the order of the finite elements increases. Numerical tests also showed that the higher order finite element method is much efficient than lower order finite element method or finite difference method.

Tube hydroforming is nowadays widely used in automotive and aircraft industries. The technology presents strong advantages when compared to conventional stamping forming. Some of them include: less parts in process and, therefore, less operations and lower costs; the use of more complex geometries, with implications on reducing weight and increasing mechanical properties; the need for fewer secondary operations; the obtaining of reduced dimensional variations; better surface finish due to the reduced friction and, finally, reduced scrap [1]. Nevertheless, the process also has some disadvantages, such as: the larger cycle time for some components; the need for relatively expensive equipment and the lack of extensive knowledge base for both process and tool designs. In this sense, the use of a reliable computational-based design tool can be highly desirable. Tailor-welded blank processes, on the other hand, consist on two or more sheets that have been welded together in a single plane prior to forming [2]. The sheets can be identical or, alternatively, can have different thickness, mechanical properties or surface coatings, leading to a versatile production process, from the mechanical properties standpoint. This type of blanks has several advantages, namely the low cost; the reduction in part weight and the flexibility in mass production. Numerical investigations of tubular hydroforming processes, starting from tailor-welded tubes based on one butt welded joint, have been carried out [3]. The need to understand this process, mainly to reduce simulation time and to accurately predict the process, leads to the present study. For this purpose, using solid-shell finite elements [4], the inclusion of the geometry of the weld line will be studied.

New types of vehicle and track elements are presented, by means of the finite element method, to establish a model for dynamic analysis of vehicle-track-subgrade coupling systems. The associated stiffness matrix, mass matrix and damping matrix for these two types of elements are deduced. Computational software is coded with Matlab. As an application example, the dynamic behavior of track transition is investigated by the vehicle and the track elements. The influencing factors for the simulation include train speed, subgrade stiffness, and irregularity angles of track transition as well as the transition pattern. The computational results show that 1) Abrupt changes of the subgrade stiffness influence the vertical acceleration of the rail and the wheel/rail contact force, and this influence increases with increases in train speed, 2) Both the irregularity angles of the track transition and the abrupt changes of the subgrade stiffness have significant effects on rail vertical acceleration and the wheel/rail contact force, with the peak value in this situation being greater than that generated by irregularity angles of the track transition or the abrupt changes of the subgrade stiffness, 3) Train speed, abrupt changes of the subgrade stiffness, and track transition irregularity angles have a minor influence on the vertical acceleration of the vehicle due to the excellent behavior of vibration isolation resulting from the primary and the secondary suspension systems of the vehicle, 4) Transition pattern for irregularity of track transition has a significant influence on the vehicle acceleration and the wheel/rail contact force, and the cosine transition is a much better indicator than the linear transition, and 5) Influences of the directions the train is moving on the dynamic behavior of the vehicle and the track are relatively insignificant.

In finite element simulations of high speed sheet metal forming processes the contact between workpiece and forming tools has to be modeled very carefully. Several important aspects have to be taken into account. Robust and locking-free finite element formulations are required to model the sheet forming process, the die has to be considered as a deformable component, and the description of the contact constraints between workpiece and forming tools is a significant source of shortcomings in modeling. The contact and impact simulation makes high demands on the robustness of finite element formulations. For this reason finite elements with low order ansatz functions are preferred. Furthermore, they prove to be advantageous when automatic meshing tools are applied. To overcome the undesired effects of locking we work with an improved version of the innovative solid-shell concept proposed by [11]. It is based on the concept of reduced integration with hourglass stabilization. The use of this solid-shell finite element allows us to test the influence of the modeling of the die and the contact constraints in a very efficient way. An overview of so-called macro and micro deformations of forming tools in sheet metal forming simulations can be found in [8]. We show that the deformation of the die has a noticeable influence in electromagnetic sheet metal forming. However, in most commercial finite element codes taking into account elastically deformable forming tools requires a full finite element discretization of the die which leads to very high computational effort. Therefore users often assume the tools as being rigid and apply node-based spring-dashpot systems to improve the modeling of the interaction between sheet metal and die. But also in this case local interactions cannot be taken into account realistically. As a possible remedy we investigate a fully elastic description of the forming tools in combination with model reduction techniques. These significantly reduce the number of degrees-of-freedom in the finite element simulation. For this reason we present different alternatives of this technique.

The paper deals with the validation of a recently proposed hexahedral solid-shell finite element in the buckling analysis of a laminated composite plate with delaminations. The object is to study the buckling behavior of structures with delaminations using the enhanced assumed strain (EAS) solid shell element with 5, 7 and 9 parameters. The EAS three-dimensional finite element formulation presented in this paper is free from shear locking and leads to accurate results for distorted element shapes. The developed FE model is used to study the effects of some parameters in the buckling load, such as the stacking sequences, delamination size, aspect ratio, width-to-thickness ratio. The feasibility of the proposed method is confirmed by numerical examples. Results show that using hexahedral solid-shell finite element in the buckling analysis is more efficient than using the enhanced solid finite element.

Finite element (FE) modelling is a popular tool for studying human body response to blast exposure. However, blast modelling is a complex problem owing to more numerous fluid–structure interactions (FSIs) and the high–frequency loading that accompanies blast exposures. This study investigates FE mesh design for blast modelling using a sphere in a closed-ended shock tube meshed with varying element sizes using both tetrahedral and hexahedral elements. FSI was consistent for sphere-to-fluid element ratios between 0.25 and 4, and acceleration response was similar for both element types (R2 = 0.997). Tetrahedral elements were found to become increasingly volatile following shock loading, causing higher pressures and stresses than predicted with the hexahedral elements. Deviatoric stress response was dependent on the sphere mesh size (p < 0.001), while the pressure response was dependent on the shock tube mesh size (p < 0.001). The results of this study highlight the necessity for mesh sensitivity analysis in blast models.

The principle of robot based incremental sheet metal forming is based on flexible shaping by means of a freely programmable path-synchronous movement of two tools, which are operated by two industrial robots. The final shape is produced by the incremental infeed of the forming tool in depth direction and its movement along the geometry’s contour in lateral direction. The main problem during the forming process is the influence on the dimensional accuracy resulting from the compliance of the involved machine structures and the springback effects of the workpiece. The project aims to predict these deviations caused by resiliences and to carry out a compensative path planning based on this prediction. Therefore a planning tool is implemented which compensates the robot’s compliance and the springback effects of the sheet metal. Finite element analysis using a material model developed at the Institute of Applied Mechanics (IFAM) [1] has been used for the simulation of the forming process. The finite strain constitutive model combines nonlinear kinematic and isotropic hardening and is derived in a thermodynamical setting. It is based on the multiplicative split of the deformation gradient in the context of hyperelasticity. The kinematic hardening component represents a continuum extension of the classical rheological model of Armstrong–Frederick kinematic hardening which is widely adopted as capable of representing the above metal hardening effects. The major problem of low-order finite elements used to simulate thin sheet structures, such as used for the experiments, is locking, a non-physical stiffening effect. Recent research focuses on the large deformation version of a new eight-node solid-shell finite element based on reduced integration with hourglass stabilization. In the solid-shell formulation developed at IFAM ([2], [3]) the enhanced assumed strain (EAS) concept as well as the assumed natural strain (ANS) concept are implemented to circumvent locking. These tools are very important to obtain a good correlation between experiment and simulation.

A method for deriving a non-singular state-space formulation based on Rubin's model for Pogo analysis is presented in this study. Because of the non-singularity, the state-space model can be directly used in frequency-domain analyses and time-domain simulations. To describe the assembly method concisely, the dynamic equations of the nine types of independent elements are described in a standard manner. The nine types of elements are divided into two classes according to characteristics of the dynamic equations. The mapping relationship between the local and global numbers of elements and nodes is obtained by numbering all of the elements and nodes. By integrating the element stiffness matrixes to obtain the total stiffness matrix used in the finite element method, the coefficient matrixes of the improved Rubin's model are assembled from the coefficient matrixes of all of the elements according to the mapping relationship. Based on the non-singular model, the time-varying simulation with the nonlinear property of the accumulator can be done conveniently by revising the state-space model. The successful application of this method to a Pogo analysis of a certain type of CZ rocket used in China verifies the correctness and efficiency of the method of this study.

Grid-based computational simulations often use hybrid unstructured grids consisting of various types of elements. Most commonly used elements are tetrahedral, pentahedral (namely, square pyramids and right triangular prisms) and hexahedral elements. Extracting isosurfaces of scalar fields defined on such hybrid unstructured grids is often done using indirect methods, such as, (a) subdividing all non-tetrahedral cells into tetrahedra and computing the triangulated isosurfaces using the marching tetrahedra algorithm, or (b) triangulating intersection points of edges of cells and an isosurface using a standard triangulation algorithm. Using the basic ideas underlying the well-established marching cubes and marching tetrahedra algorithms, which are applied to hexahedral and tetrahedral elements, respectively, we generate look-up tables for extracting isosurfaces directly from pentahedral elements. By using appropriate look-up tables, it is possible to process nearly all types of hybrid unstructured grids used in practical applications without the need for indirect methods. We construct look-up tables for square pyramidal and triangular prismatic cells with accurate topological considerations.

2 - Discretization of the Domain

Certain difficulties with the use of quadrilateral or hexahedral finite elements are related to mesh refinement and to element compatibility and quality after refinement. In this paper, special refinement elements are presented that make possible connecting two special elements to one edge of an 8 node quadrilateral element (2D). The main idea in refinement elements is to place some midside nodes outside the element area and to modify element shape functions in order to maintain continuity at a refinement edge. Special refinement elements allow to adaptively refine mesh in such a way that it fits the quadtree data structure. In the three-dimensional case, hexahedral elements with 20 nodes are employed. Placement of some nodes outside the element volume allows us to create a compatible octree refinement scheme.

It has been recognised that numerical computations of magnetic fields by the finite-element method may require new types of elements, whose degrees of freedom are not field values at mesh nodes, but other field-related quantities like e.g. circulations along edges of the mesh. A rationale for the use of these special ‘mixed’ elements can be obtained if one cxpresses basic equations in terms of differential forms, instead of vector fields. The paper gives an elementary introduction to this point of view, presents Whitney forms (the mixed finite elements alluded to), and sketches two numerical methods (dual, in some sense), for eddy-current studies, based on these elements.

Multigroup application of the anisotropic FEM code FELTRAN to one, two, three-dimensions and R- Z problems

To analyse the behaviour of thin structures of dielectric elastomer (DE) material a solid shell finite element is presented. The main characteristics of DEs are a non‐linear hyper elastic behaviour, the quasi‐incompressibility, and the ability to transform electric energy into mechanical work. Applying a voltage to thin DE structures may produce large elongation strains of 120‐380%. These large strains, the efficient electro‐mechanical coupling, and the light weight make DEs very attractive for the usage in actuators. Thus, there is a need for detailed research. With respect to the electro‐mechanical coupling a constitutive model is presented. An electric stress tensor and a total stress tensor are introduced by considering the electrical body force and couple in the balance of linear momentum and angular momentum, respectively. The governing equations are derived and embedded in the solid shell formulation. The element formulation is based on a Hu‐Washizu mixed variational principle using six independent fields: displacements, electric potential, strains, electric field, mechanical stresses, and dielectric displacements. It allows large deformations and accounts for physical nonlinearities to capture two of the main characteristics of DEs. The shell element could be applied for the modelling of arbitrary curved thin structures. The ability of the present element formulation is demonstrated in several examples. (© 2011 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

This study is focusing on the finite element simulation of low velocity impact applied to glass fiber-reinforced composites. Elasto-visco-plasticity is adopted taking into account strain rate effect, with isotropic hardening. To avoid numerical locking, hexahedral solid shell finite element SS is developed. The formulation is based on the mixed Assumed Natural Strain (ANS) and the Enhanced Assumed Strain (EAS) methods. A geometrical non linear condition is applied. The finite element is implemented in UEL subroutine via the commercial software ABAQUS/Standard. The numerical element proved its efficiency comparing to conventional element and previous experimental results in (Mars et al. 2018). Numerical simulations with solid element are conducted using Abaqus/standard via user material UMAT subroutine. In terms of computation time, the solid shell element is less time consuming when comparing to conventional solid element. This is due to the fact that a single element through the thickness direction is adopted. Element efficiency is assessed for various fibers contents.