Arbitrary lagrangian-eulerian petrov-galerkin finite elements for nonlinear continua

This paper proposes a simple method for deriving the geometric stiffness matrix (GSM) of a three‐node triangular plate element (TPE). It is found that when the GSM of the element is combined into the global one of the structure, this structural stiffness matrix becomes symmetric and satisfies both the rigid body rule and incremental force and moment equilibrium (IFE) conditions, which are basically two fundamental conditions for analysis of mechanics. The former condition has been widely used in the community of mechanics; while the latter one, to our best knowledge, has never been considered. Advantages with the GSM derived are that derivations only need simple matrix operations without cumbersome non‐linear virtual strain energy derivations and tedious numerical integrations and more appealingly, this derived GSM can be explicitly given for applications. In addition, based on IFE and the rigid body rule conditions, a reasonable GSM for the three‐node TPE must be asymmetric; however, an asymmetric matrix usually gives rise to tedious numerical calculation especially in geometrically nonlinear problems and further, greatly influences computation efficiency. Fortunately, the skew‐symmetric parts of the derived GSM can be canceled out once they are merged into the global stiffness matrix of the structure. In this regard, this structural stiffness matrix becomes a symmetric one and thus enhances its effectiveness. Finally, several examples are provided for validating the robustness of the derived GSM.

Abstract. Material Extrusion (MEX) and by extension Fused Deposition Modeling (FDM) is a popular Additive Manufacturing (AM) process used to fabricate complex parts. FDM as of recent is no longer solely utilized for prototyping parts but also used for producing functional components in industrial and research applications. The investigation of material properties of these FDM parts by experimental means is a time-consuming task. Therefore, the use of numerical simulation methods is highly required. Since 1993, various Finite Element Analysis (FEA) models are found in literature attempting to effectively simulate FDM parts utilizing many formulations, each with their pros and cons. The present study aims to compare Lagrangian and Arbitrary Lagrangian-Eulerian (ALE) finite element formulations in simulating tensile loading for FDM parts. The efficiency and precision of the aforementioned methods is evaluated in the numerical simulation of the tensile loading of an ASTM D638 standard geometry ABS specimen utilizing both ABAQUS/Explicit and ABAQUS/Standard. Utilizing a paper from the literature for experimental validation, this study additionally provides insight into explicit and implicit models’ computational efficiency, focusing on the advantages of explicit models for this application. The effects of mesh element type and size on results are also studied for each method. Based on these results, some useful guidelines for selecting the most suitable model of tensile loaded FDM parts are presented.

This paper describes a new finite element scheme for the analysis of instability phenomena of arbitrary thin shells. A computationally efficient procedure is proposed for calculating the non‐linear stiffness and tangential stiffness matrices for a doubly‐curved quadrilateral element defined by co‐ordinate lines. The essential feature is the explicit addition of the non‐linear terms into the rigid‐body motion of the element. Thus the non‐linear and tangential element stiffness matrices can easily be generated by transforming the generalized element stiffness matrix for linear analysis, and the non‐linear terms of these matrices are separated into a number of component terms multiplied by the rigid‐body rotations. These component terms can be stored permanently and used to calculate efficiently the non‐linear and tangential stiffness matrices at each iteration. Illustrative examples are presented which confirm the validity of the present approach in the analysis of instability phenomena of thin plates and shells.

A simplified geometric stiffness in stability analysis of thin-walled structures by the finite element method

This paper presents a complete derivation and implementation of the Arbitrary Lagrangian Eulerian (ALE) formulation for the simulation of large deformation quasi-static and dynamic problems. While most of the previous work done on ALE for dynamic applications was mainly based on operator split and explicit calculations, this work derives the quasi-static and dynamic ALE equations using a fully coupled implicit approach. Full expression for the ALE virtual work equations and finite element matrices are given. Time integration relations for the dynamic equations are also derived. Several quasi-static and dynamic large deformation applications are solved and presented.

Kinematic analysis of linkages based in finite elements and the geometric stiffness matrix

Geometric nonlinear static analysis of structural systems with cable elements is carried out using point based iterative procedure. In all sub systems as cable systems, constituted for each node having at least one degree of freedom that is idealized by finite elements, successive calculations are performed. In the analysis part, based on finite element displacement method, to the maximum number of unknown displacements required for each sub-system calculation is limited with three. Tangent stiffness matrix, including pre-stressed internal forces as well as varying geometries with respect to different external force applications, is utilized. The convergence procedure is adapted into the method to prevent excessive displacements through the calculations. In the present study, a computer program has been developed to present a very effective calculation method. Different numerical applications have been considered and the results were compared with the literature results. Key words: Cable systems, nonlinear analysis, point based iterative procedure, convergence procedure.

A novel affine decomposition strategy for the finite element linear and geometric stiffness matrices of a fiber composite plate structure is presented here. The main goal of this decomposition is to express the stiffness matrices as an affine summation of terms formed by parameter-independent matrices and parameter-dependent scalars. Such a grouping has applications in improving efficient finite element analysis and improved performance of metaheuristic optimization algorithms used in structural optimization. The linear stiffness matrix is decomposed using lamination parameters and invariant matrices from classical lamination theory. The decomposition of the geometric stiffness matrix is more complicated due to the involvement of the in-plane load-induced transverse stiffness. First, the static response of the structure is approximated using a set of projection basis vectors, and the coefficient associated with each vector has an implicit dependence on the fiber composite configuration. The design-independent components of the geometric stiffness matrix are computed in terms of the projection basis vectors of the static response, and the coefficients are computed on the fly using least squares approximation. The application of these decompositions in a parametric model order reduction-based two-step optimization framework is demonstrated. The proposed approach resulted in a significant reduction in the time and resource requirement needed to carry out this optimization.

This paper presents the finite rotation exact geometry (EG) 12-node solid-shell element with 36 displacement degrees of freedom. The term ‘EG’ reflects the fact that coefficients of the first and second fundamental forms of the reference surface and Christoffel symbols are taken exactly at each element node. The finite element formulation developed is based on the 9-parameter shell model by employing a new concept of sampling surfaces (S-surfaces) inside the shell body. We introduce three S-surfaces, namely, bottom, middle and top, and choose nine displacements of these surfaces as fundamental shell unknowns. Such choice allows one to represent the finite rotation higher order EG solid-shell element formulation in a very compact form and to derive the strain–displacement relationships, which are objective, that is, invariant under arbitrarily large rigid-body shell motions in convected curvilinear coordinates. The tangent stiffness matrix is evaluated by using 3D analytical integration and the explicit presentation of this matrix is given. The latter is unusual for the non-linear EG shell element formulation. Copyright © 2011 John Wiley & Sons, Ltd.

A computational framework for fluid–rigid body interaction: Finite element formulation and applications

Thanks to their properties of superelasticity and shape memory effects, SMA wires are often considered for the design of damping devices or actuators coupled to pulleys. In order to dimension such devices for commercialization, it is necessary to develop reliable, robust and fast numerical tools. To this end, simplifying the management of the wire/pulley contact using an Arbitrary Lagrangian–Eulerian (ALE) formulation is an original and effective solution. In this work, the wires are modeled by three-dimensional truss thermo-mechanical finite elements associated with a superelastic law. For each node, the curvilinear abscissa is used as an additional degree of freedom in order to deal with material flow from one element to another. After presenting the ALE formalism and the necessary precautions (advection of variables), as well as the material behavior and thermo-mechanical contact laws at the pulleys, elementary validation tests are studied. Finally, two devices taken from the literature are modeled using the ALE formalism. The results obtained demonstrate the relevance and effectiveness of the approach adopted, as well as the importance of not neglecting friction at the pulleys and thermal effects on the material mechanical response.

An improved ALE and CBS-based finite element algorithm for analyzing flows around forced oscillating bodies

Instability of laminated composite thin-walled open-section beams

Large amplitude free flexural vibrations of thin plates of arbitrary shape

An arbitrary Lagrangian–Eulerian finite element formulation for a poroelasticity problem stemming from mixture theory