Academic literature on the topic 'Nodal integration technique'

In this paper, a nodal integration method (NIM) is presented to deal with the static and dynamic problems of Timoshenko beam. In the present method, linear-shape functions are employed to approximate the displacement field, and smoothing domains based on the nodes are further formed for computing the stiffness matrix. Through a smoothing operation, the shear locking is effectively avoided and the computation gets much simpler. For static problems, the upper bounds for a set of benchmark examples are obtained by nodal integration. For dynamic problems, while keeping the shear stiffness matrix the same as NIM, integration based on elements is adopted to construct the bending stiffness matrix to improve the stability and diminish singular modes caused by pure nodal integration. Results computed in this way prove to be much better than pure nodal integration method for free vibration and forced vibration problems. Numerical examples indicate that very accurate results can be obtained when a reasonable number of nodes is used. Both computational efficiency and accuracy are achieved by above formulations.

FEM implicit formulation shows specific limitations in processes such as cutting, where large deformation results in a heavy mesh distortion. Powerful rezoning-remeshing algorithms strongly reduce the effects of such a limitation but the computational times are significantly increased and additional errors are introduced. Nodal Integration is a recently introduced technique that allows finite element method to provide more reliable results when mesh becomes distorted in traditional FEMs. Furthermore, volumetric locking phenomenon seems to be avoided by using this integration technique instead of other methods, such as the coupled formulations. In this paper, a comparison between a “classical” FEM simulation and the Nodal Integration one is carried out taking into account a simple orthogonal cutting process.

A new efficient updated-Lagrangian strategy for numerical simulations of material forming processes is presented in this work. The basic ingredients are the in-plane-out-of-plane PGD-based decomposition and the use of a robust numerical integration technique (the Stabilized Conforming Nodal Integration). This strategy is of general purpose, although it is especially well suited for plateshape geometries. This paper is devoted to show the feasibility of the technique through some simple numerical examples.

Esophageal cancer is an uncommon but highly lethal disease. Surgical resection is the gold standard of treatment for early-stage disease. Traditional surgical approach entailed significant convalescence, hospital stay, and morbidity and mortality. Transhiatal esophagectomy (THE) involves blind dissection of the esophagus with minimal mediastinal lymphadenectomy. Integration of robotic surgery is an alternate platform for minimally invasive approach while maintaining safety and following oncologic principles. We review our technique for minimally invasive THE using robotic technology, demonstrating the safety and efficacy of robotic technology surgery. We present a retrospective review of a single surgeon's data of patients treated with robotic-assisted THE, with a chart review to evaluate pathology, adequacy of surgical resection, nodal harvest, and perioperative course. Robotic THE (rTHE) shows promise as a valid option for esophageal resection, including premalignant and advanced stages of cancer. Adequate transhiatal mediastinal nodal resection can be performed with the robot.

This paper presents a thin plate formulation with nodal integration for bending analysis using three-node triangular cells and linear interpolation functions. The formulation was based on the classic thin plate theory, in which only deflection field was required and dealt with as the field variables. They were assumed to be piecewisely linear and expressed using a set of three-node triangular cells. Based on each node, the integration domain has been further derived, where the curvature in the domain was computed using a gradient smoothing technique (GST). As a result, the curvature in each integration domain is a constant whereby the deflection is compatible in the whole problem domain. The generalized smoothed Galerkin weak form is then used to create the discretized system equations where the system stiffness is obtained using simple summation operation. The essential rotational boundary conditions are imposed in the process of constructing the curvature field in conjunction with imposing the translational boundary conditions in the same way as undertaken in the standard FEM. A number of numerical examples were studied using the present formulation, including both static and free vibration analyses. The numerical results were compared with the reference ones together with those shown in the state-of-art literatures published. Very good accuracy has been achieved using the present method.

The shallow water equations (SWE) is often simulated by using Eulerian descriptions. These phenomena may give rise to strong gradients and lead to large distortion of grids meshes. Hence classical finite elements methods may fall in simulating such problems. In this paper we present a meshless method, based on the natural element nethod (NEM). In a geometrical domain of a cloud of nodes, NEM uses the Voronoi cells and then its dual, namely Delaunay triangulation. Its main advantage lies in shape function of the natural neighbour interpolation, such that the position of natural neighbours is enough to its construction. To avoid the nonlinear term, the time material derivative term is discretized by a Lagrangian procedure. We also used an appropriate nodal integration technique to estimate integrals related to the diffusion, pressure and Coriolis terms because NEM shape functions are not polynomials and they are rational. For the diffusion term, the method of stabilized conforming nodal integration (SCNI) is proposed while for pressure and Coriolis terms a geometrical method will transform the integration over the cells domain to the integration over the edges. The method was successfully used to simulate dam-break flows by solving the fully 2D shallow water equations (SWE) by using an implicit scheme under a transient flow.

De nombreux procédés de fabrication thermomécanique comme le laminage, le soudage ou encore l’usinage mettent en jeu soit des sollicitations mobiles par rapport à la matière fixe, soit de la matière mobile par rapport à des sollicitations fixes. Dans tous les cas, après un régime transitoire en général assez court, les champs thermiques, métallurgiques et mécaniques associés à ces procédés atteignent un état stationnaire. La recherche de ces états stationnaires à l’aide de la méthode des éléments finis classique nécessite de mettre en œuvre des modèles complexes et couteux où les sollicitations se déplacent par rapport à la matière (ou l’inverse). La recherche directe des états stationnaires a fait l’objet de nombreux travaux de recherche ces trente dernières années. Des méthodes sont aujourd’hui disponibles et pour certaines sont proposées dans des codes de calcul du commerce. Ainsi, une option de calcul dite repère mobile proposée par différents auteurs est disponible dans le logiciel SYSWELD. Cette méthode permet de calculer les états thermique, métallurgique et mécanique stationnaires associés à un procédé de soudage, en résolvant un problème de diffusion-convection en thermique et en intégrant, en mécanique, les équations constitutives du comportement du matériau le long des lignes de courant. Si cette méthode a été utilisée avec succès dans de nombreuses applications, elle présente néanmoins quelques limitations. Ainsi le maillage doit être structuré et la convergence des calculs est en général assez lente. Nous proposons dans cette thèse de résoudre le problème mécanique dans un repère lié aux sollicitations, en nous appuyant sur une méthode de calcul par éléments finis reposant sur l’intégration nodale et la technique SCNI (Stabilized Conforming Numerical Integration). Cette méthode permet l’utilisation de maillages en tétraèdres (ou triangles en 2D) sans rencontrer de problème de verrouillage volumique résultant de l’incompressibilité plastique associée au critère de plasticité de von Mises. Plutôt que de rechercher directement l’état stationnaire, l’idée générale est ici de construire l’état stationnaire à partir d’une analyse transitoire en faisant entrer pas à pas la matière par l’amont et en la faisant sortir par l’aval d’un maillage fixe par rapport aux sollicitations et de taille limitée. L’état stationnaire n’est donc atteint qu’au bout d’un certain temps d’analyse. Après une introduction générale (Chapitre 1) et un état de l’art sur les méthodes existantes (Chapitre 2), nous présentons une approche de simulation du mouvement de matière dans le cadre de la méthode des éléments finis classique sur un problème de soudage (Chapitre 3). Nous y proposons également des conditions aux limites thermiques pertinentes pour calculer directement la distribution de températures en régime stationnaire. La méthode des éléments finis reposant sur l’intégration nodale est ensuite décrite au Chapitre 4. Les avantages et inconvénients de la méthode sont discutés. La méthode est validée sur une application en grandes déformations élastoplastiques, un problème de flexion et une simulation thermomécanique de soudage. La méthode des éléments finis reposant sur l’intégration nodale est alors développée pour prendre en compte un mouvement de matière (Chapitre 5). Trois types de mouvement sont considérés : en translation, circulaire et en hélice. Différentes méthodes de transport de champ sont abordées et discutées ainsi que le couplage thermomécanique. Des perspectives à ce travail sont proposées au Chapitre 6. Les perspectives envisagées visent d’une part à améliorer la méthode proposée et d’autre part, à développer la méthode pour simuler d’autres procédés. Une première application de la méthode à la simulation de la coupe orthogonale y est présentée
In the numerous thermomechanical manufacturing processes such as rolling, welding, or even machining involve either moving loads with respect to the fixed material or moving material with respect to fixed loads. In all cases, after a transient regime which is generally quite short, the thermal, metallurgical, and mechanical fields associated with these processes reach a steady state. The search for these stationary states using the classical finite element method requires the implementation of complex and expensive models where the loads move with respect to the material (or vice versa). The steady-state simulation in one increment has been the subject of much researches over the past thirty years. Methods are now available and some are integrated into calculation codes commercial. Thus, a so-called Moving Reference Frame method proposed by various authors is available in the SYSWELD software. This method makes it possible to calculate the steady-state of thermal, metallurgical, and mechanical states associated with a welding process, by solving a thermal diffusion-convection problem in thermal-metallurgy and by integrating, in mechanics, the constitutive equations of the material along the streamline. Moreover, this method has been used successfully in many applications, it nevertheless has some limitations. Thus the mesh must be structured and the convergence of computations is generally quite slow. In this thesis, we propose to solve the mechanical problem in a frame linked to the solicitations, by relying on a finite element calculation method based on nodal integration and the SCNI (Stabilized Conforming Numerical Integration) technique. This method allows the use of tetrahedron meshes (or 2D triangles) without encountering a locking problem resulting from the plastic incompressibility associated with the von Mises plasticity criterion. Rather than directly calculating the steady-state, the general idea here is to construct the steady-state from a transient analysis by bringing material step by step upstream and by making it exit downstream of a fixed mesh related to the solicitations and of the limited mesh size. The steady-state is therefore only achieved after certain steps of analysis. Apart from a general introduction (Chapter 1) and a state of the art on the existing methods (Chapter 2), we present an approach of simulation of the movement of material within the framework of the classical finite element method on a welding problem (Chapter 3). We also provide relevant thermal boundary conditions for directly calculating the steady-state of temperature distribution. The finite element method based on the nodal integration technique is then described in Chapter 4. The advantages and disadvantages of the method are discussed. The nodal-integration-based finite element is validated by comparing its simulation results with classical finite element methods in large elastoplastic strains, a bending problem, and a thermomechanical simulation of welding. The nodal-integration-based finite element is then developed and applied to simulate material motion (Chapter 5). Three types of movement are considered: translational, circular, and helical. Different methods of field transport are approached and discussed as well as thermomechanical coupling. Perspectives for this work are presented in Chapter 6. The envisaged perspectives aim, on the one hand, to improve the proposed method and on the other hand, to develop the method to simulate other processes. A first application of the material motion method to the simulation of the orthogonal cut is presented there

Diese Arbeit widmet sich der Simulation von elektrischen/elektronischen Schaltungen welche um elektromagnetische Bauelemente erweitert werden. Im Fokus stehen unterschiedliche Kopplungen der Schaltungsgleichungen, modelliert mit der modifizierten Knotenanalyse, und den elektromagnetischen Bauelementen mit deren verfeinerten Modell basierend auf den vollen Maxwell-Gleichungen in der Lorenz-geeichten A-V Formulierung welche durch Finite-Integrations-Technik räumlich diskretisiert werden. Eine numerische Analyse erweitert die topologischen Kriterien für den Index der resultierenden differential-algebraischen Gleichungen, wie sie bereits in anderen Arbeiten mit ähnlichen Feld/Schaltkreis-Kopplungen hergeleitet wurden. Für die Simulation werden sowohl ein monolithischer Ansatz als auch Waveform-Relaxationsmethoden untersucht. Im Mittelpunkt stehen dabei Zeitintegration, Skalierungsmethoden, strukturelle Eigenschaften und ein hybride Ansatz zur Lösung der zugrundeliegenden linearen Gleichungssysteme welcher den Einsatz spezialisierter Löser für die jeweiligen Teilsysteme erlaubt. Da die vollen Maxwell-Gleichungen zusätzliche Ableitungen in der Kopplungsstruktur verursachen, sind bisher existierende Konvergenzaussagen für die Waveform-Relaxation von gekoppelten differential-algebraischen Gleichungen nicht anwendbar und motivieren eine neue Konvergenzanalyse. Auf dieser Analyse aufbauend werden hinreichende topologische Kriterien entwickelt, welche eine Konvergenz von Gauß-Seidel- und Jacobi-artigen Waveform-Relaxationen für die gekoppelten Systeme garantieren. Schließlich werden numerische Benchmarks zur Verfügung gestellt, um die eingeführten Methoden und Theoreme dieser Abhandlung zu unterstützen.
This work is devoted to the simulation of electrical/electronic circuits incorporating electromagnetic devices. The focus is on different couplings of the circuit equations, modeled with the modified nodal analysis, and the electromagnetic devices with their refined model based on full-wave Maxwell's equations in Lorenz gauged A-V formulation which are spatially discretized by the finite integration technique. A numerical analysis extends the topological criteria for the index of the resulting differential-algebraic equations, as already derived in other works with similar field/circuit couplings. For the simulation, both a monolithic approach and waveform relaxation methods are investigated. The focus is on time integration, scaling methods, structural properties and a hybrid approach to solve the underlying linear systems of equations with the use of specialized solvers for the respective subsystems. Since the full-Maxwell approach causes additional derivatives in the coupling structure, previously existing convergence statements for the waveform relaxation of coupled differential-algebraic equations are not applicable and motivate a new convergence analysis. Based on this analysis, sufficient topological criteria are developed which guarantee convergence of Gauss-Seidel and Jacobi type waveform relaxation schemes for introduced coupled systems. Finally, numerical benchmarks are provided to support the introduced methods and theorems of this treatise.

Liquid-solid phase transition is accompanied by a latent heat release, both in isothermal and non-isothermal phase transformations. The latent heat and the discontinuous phase change function increase the difficulty of obtaining a solution for the Fourier heat conduction equation. Celentano et al. [Int. J. Numer. Meth. Eng. 37(20), 1994] proposed a temperature-based finite element model for solving multidimensional transient heat conduction involving phase change. The present work addresses the computational aspects of the Celentano et al. model. The importance of a line search algorithm for improving the convergence of the Newton-Raphson iterations are explained in detail. While performing the iterations in this kind of fixed domain methods, the phase front moves back and forth fictitiously. The introduced phase change matrix handles the latent effect efficiently. The phase fractions are evaluated at the integration points instead of the nodal points. Several numerical examples are presented and the benefits and difficulties of the solution technique are elaborately discussed.

An analytic basis function expansion nodal method for directly solving the two-group neutron diffusion equation in the triangular geometry is proposed in the present paper. In this method, the distribution of neutron flux is expanded by a set of analytic basis functions. The diffusion equation is satisfied at any point in a triangular node for each group assuming that the flux within a node is flat. No transverse integration is needed. To improve the nodal coupling relations and computation accuracy, nodes are coupled with each other fulfilling both the zero- and first-order partial neutron current moments across all the three interface of the triangle mesh at the same time. Coordinate conversion is used to transform arbitrary triangle into regular triangle in order to simplify the derivation. A new sweeping scheme is developed for the triangular mesh and the response matrix technique was used to solve the nodal diffusion equation iteratively. Based on the proposed model, the code ABFEM-T is developed. Validation of code for accuracy and efficiency are carried out by calculating both rectangular and hexagonal assembly benchmark problems. Numerical results for the series of benchmark problems show that both the multiplication factor and nodal power distribution are predicted accurately. Therefore this method can be used for solving neutron diffusion problems in complex unstructured geometry.

Abstract A methodology for the automatic mesh generation of triangular and quadrilateral finite element discretizations for two-dimensional elasticity problems is proposed. The methodology is based on: i) an h-adaptive process with powerful mesh generator facilities capable of achieving meshes of specified density, ii) a general stress recovery technique developed for determining the element solutions at the nodes, and iii) an a posteriori error estimation. The h-adaptive process used is based on a complete mesh regeneration procedure which is guided by specified mesh requirements such as geometry definitions, boundary conditions, and space node functions to achieve an optimal refinement. This optimality condition is, as established by Zienkiewicz and Zhu, the mesh refinement with the least number of elements that yields a uniform strain energy norm error distribution in all elements. In the stress recovery process, the nodal values are assumed to belong to a polynomial expansion of the same complete order in the interpolation function basis used, which is valid over all elements adjoining a particular node. A least-squares fit of superconvergent sampling points existing in the path is used to obtain the recovered nodal point parameters for each element. These parameters are averaged to all elements adjoining the node of interest. The technique is simple and cost-effective, and the recovered nodal values of derivatives are superconvergent at the Gauss integration points, which are used as sampling points for quadrilateral elements. This condition is also achieved when centroid and mid-side points are used for triangular elements. The error estimation is done evaluating differences between the post-processed stress gradients and those from the finite element solutions. The energy error norm associated with stress field differences and the finite element predicted strain energy gives an effective error estimate which can be used for comparison with the process tolerance. The technique has been implemented and allows for a fully automatic numerical analysis under a specified global energy error norm. Numerical tests conducted with various planar element formulations illustrate that the proposed technique converges in fewer steps than with previous methods of adaptive mesh refinement.

Abstract In this paper, a generalized finite element method (GFEM) with local gradient smoothed approximation (LGS-GFEM) using triangular meshes is proposed. The displacement field function of LGS-GFEM consists of the finite element shape function and the node displacement function. In order to obtain the nodal displacement function, the second order Taylor expansion is considered. The derivative term in Taylor expansion is obtained by using gradient smoothed technique in a smoothed domain. The displacement in smoothed operation is interpolated by polynomial basis function and radial basis function. Two kinds of integration schemes are considered, i.e. LGS-GFEM-I and LGS-GFEM-II respectively. The smoothed composite shape function of LGS-GFEM retains the ideal Kronecker property of the finite element shape function. Besides, the proposed LGS-GFEM has some other important properties such as no extra DOFs, linear independent, etc. The superiority of LGS-GFEM including high accuracy, rapid error convergence and temporal stability, is demonstrated by two representative numerical examples of static and free vibration, and compared with the classical finite element of triangular (FEM-T3) and quadrilateral (FEM-Q4) elements.

The sloshing analysis of liquid storage tanks by the finite element method is basically categorized into two approaches, Lagrangian approach and Eulerian approach. In the Lagragian approach, the behavior of the fluid is expressed in terms of the displacements at nodal points. The advantage of the Lagragian method is that the computer code can be easily developed to modify an existing structural analysis code. The disadvantage is that some spurious modes are included in the vibration modes. The Lagrangian method is widely used in two- and three-dimensional problems. On the other hand, it has not been reported its applicability to the axisymmetric problem. This paper presents the applicability of the Lagragian method to the axisymmetric sloshing problem. The eigenvalue of an elemental stiffness matrix is analyzed in order to investigate the characteristics of the rotational stiffness to the compressibility of the fluid. As a result, this method is found to be difficult to apply to the axisymmetric problem if the equation of motion is directly solved using time integration. However, it gives the highly precise response solutions if the only sloshing modes are taken out and the modal analysis technique is used.

In order to improve the situation that the design of microactuator is mostly based on the intuition and experience of researchers, the method of continuum topology optimization using the nodal density is introduced to the conceptual design of microactuator. This new method can ensure C0 continuity of density field in a fixed design domain. The ratio of mutual energy to strain energy of the mechanism is regarded as the objective function, where, the mutual energy and strain energy describe the kinematic function and structural function of microactuator respectively. The final configuration of microactuator is decided on the guide of conceptual design combined with the given working conditions. The finite element method is applied to analysis the transmission ratio and clamping force of microactuator. The prototype of the microactuator is fabricated by using micro-electroforming and SU-8 photolithography techniques and the displacement of the micro actuator is measured by using the stereo vision microscopy. The experimental results show that the properties of the micro actuator can satisfy the designing demands. This topological optimization method based on nodal density plays an important role in guiding the structure design of micro actuator.

In the last decade Vilhena and coworkers10 reported an analytical solution to the two-dimensional nodal discrete-ordinates approximations of the neutron transport equation in a convex domain. The key feature of these works was the application of the combined collocation method of the angular variable and nodal approach in the spatial variables. By nodal approach we mean the transverse integration of the SN equations. This procedure leads to a set of one-dimensional SN equations for the average angular fluxes in the variables x and y. These equations were solved by the old version of the LTSN method9, which consists in the application of the Laplace transform to the set of nodal SN equations and solution of the resulting linear system by symbolic computation. It is important to recall that this procedure allow us to increase N the order of SN up to 16. To overcome this drawback we step forward performing a spectral painstaking analysis of the nodal SN equations for N up to 16 and we begin the convergence of the SN nodal equations defining an error for the angular flux and estimating the error in terms of the truncation error of the quadrature approximations of the integral term. Furthermore, we compare numerical results of this approach with those of other techniques used to solve the two-dimensional discrete approximations of the neutron transport equation6.

Almost all industrial products are assembled from multiple parts, and this is true for all sizes of products. As an example, a nuclear facility is a large structure consisting of more than 10 million components. This paper discusses a method to analyze an assembly by gathering data on its component parts. Gathered data on component may identify ill conditioned meshes for connecting surfaces between components. These ill meshes are typified by nodal point disagreement in finite element discretization. A technique to resolve inconsistencies in data among the components is developed. By using this technique, structural analysis for an assembly can be carried out, and results can be obtained by the use of supercomputers, such as the K computer. Numerical results are discussed for components of the High Temperature Engineering Test Reactor of the Japan Atomic Energy Agency.

ASME Code stress assessment of pressure vessels in the power generation industry is usually done by finite element analysis using one of the two approaches. In the first, “shell-element” approach, vessels are modeled out of shell elements; primary plus bending and primary plus secondary stresses are taken directly from the finite element analysis results and the alternating stresses are based on primary plus secondary stresses prorated by respective stress concentration factors. The strength of the “shell-element” approach is its simplicity; its weakness is problematic modeling of the stress concentration and some modeling difficulties (varying wall thickness, nozzle/vessel connectivity, pressure applied to the mid-surface instead of to the inner surface.) In the second, “solid-element” approach, vessels are modeled out of solid elements; “linearized” stresses can not be taken directly from the finite element analysis results, first they must be linearized, and only then, can be compared against their allowable counterparts; the alternating stresses can be based directly on the outer/inner-surface-node-stresses, provided that the mesh of the model is fine enough to account for the stress concentration effect. The strength of the “solid-element” approach is its high accuracy; its weakness is the time consuming, sometimes ambiguous, stress linearization process. This paper proposes a modification of the “solid-element” approach, in which the time consuming linearization process is replaced by a modification of the original model. To do so, a vessel must be modeled out of quadratic 20 node solid elements; the mesh density of the model (on its surface and through thickness) must be adequate for stress concentration representation and the mesh lines in the thickness direction must be more or less normal to the surfaces. The results from this original model can be taken directly for fatigue evaluation. To obtain the “linearized” stresses the original model must be slightly modified, specifically the number of elements through thickness must be reduced to one, and the reduced integration technique is recommended. For such a modified model, the nodal stresses are equivalent to the “linearized stresses” of the original model. The equivalence is discussed on a model of a circular nozzle attached to a cylindrical vessel. The vessel loads are pressure and thermal expansion.

A couple of issues are identified in the process to embed absolute nodal coordinate formulation (ANCF) flexible bodies in an existing multibody dynamics code. (1) The generalized coordinates of ANCF must be solved together with those of the rest of the mechanism in a combined system of the equations of motion. (2) The various constraints, joints, and forces elements supported in the multibody dynamics code must be extended to the ANCF flexible bodies without major code restructuring. This paper describes two novel techniques that were devised to solve these issues. The first is the idea of interface triad. We will demonstrate how to construct the interface triad such that all exiting constraints, joints, and forces elements are automatically supported. The second idea is to represent the equations of motion of the ANCF body as a user-defined subroutine element representing a set of implicit general state equations subroutine (GSESUB). By treating each ANCF body modularly as a user-defined subroutine, not only all existing integration options of its host solver, e.g., HHT or DAE index-1, 2, and 3, etc., are automatically supported, but also the existing features such as parallel computing and sparse matrix solution of the existing multibody dynamics software are supported with minimum programming. Numerical examples are presented to demonstrate the efficiency and the success of these two techniques.