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The finite element quasi‐static analysis of elastoplastic systems is studied by making use of a generalized variable approach for the spatial discretization and a generalized mid‐point rule for the time integration. Both the classical form of the constitutive law and the convex analysis formulation are presented. The relation between the mid‐point time integration and the extremal path theory is discussed. Extremal properties for the finite‐step solution are formulated.

This paper addresses the loading/unloading conditions of thediscrete initial—value problem of plastic flow at infinitesimal deformations. As in the continuum problem, it is established that the strain—space formulation of the loading conditions is primary. Generalized trapezoidal and mid‐point rules are discussed. The loading conditions established for the general non‐associated flow problem are shown to naturally reduce to well‐known inequalities for flow rules obeying normality.

Various stress return algorithms in elastoplastic analyses using the finite element method require the evaluation of the contact (or penetration) stress state (Figure 1), defining the transition from elastic to elastoplastic behaviour. Various iterative schemes are commonly used to evaluate contact stress state with a great degree of precision, as subsequent analysis process (forward Euler, mid‐point rule stress return scheme) is greatly affected by the evaluation of the contact stress state, as has been stressed by several authors.

This paper presents a set of Mathematica modules that organizes numerical integration rules considered useful for finite element work. Seven regions are considered: line segments, triangles, quadrilaterals, tetrahedral, wedges, pyramids and hexahedra. Information can be returned in floating‐point (numerical) form, or in exact symbolic form. The latter is useful for computer‐algebra aided FEM work that carries along symbolic variables. A few quadrature rules were extracted from sources in the FEM and computational mathematics literature, and placed in symbolic form using Mathematica to generate own code. A larger class of formulas, previously known only numerically, were directly obtained through symbolic computations. Some unpublished non‐product rules for pyramid regions were found and included in the collection. For certain regions: quadrilaterals, wedges and hexahedra, only product rules were included to economize programming. The collection embodies most FEM‐useful formulas of low and moderate order for the seven regions noted above. Some gaps as regard region geometries and omission of non‐product rules are noted in the conclusions. The collection may be used “as is” in support of symbolic FEM work thus avoiding contamination with floating arithmetic that precludes simplification. It can also be used as generator for low‐level floating‐point code modules in Fortran or C. Floating point accuracy can be selected arbitrarily. No similar modular collection applicable to a range of FEM work, whether symbolic or numeric, has been published before.

The numerical simulation of metal forming processes follows a highly non‐linear analysis where general aspects such as elastoplasticity, finite deformation and contact mechanics are combined. Approximated solutions obtained by finite element techniques require strong computational effort, that contradicts the need of interactive industrial applications. The first part of the work deals with the description of the main elements of the formulation, with attention to mathematical modelling and the approximating algorithms in the incremental iterative frame of non‐linear analysis, ending with the results obtained in hot rolling simulation. The second part is dedicated to computational efficiency analysis and the presentation of the related methods and results obtained in this work.

An iterative algorithm is described to compute Schwarz‐Christoffel transformations which map the upper half of a complex plane into the interior of a polygon in another complex plane. An efficient method of numerically integrating the S‐C integral over the singularities is presented. The algorithm is easily programmable in FORTRAN. Convergence rate is high and accuracy is excellent. Examples are provided and wherever possible, analytically obtained results are also presented for comparison. The importance of the algorithm is described and a brief comparison with some of the existing algorithms is made. Potential application of the S‐C transformation are in the solution of Laplace's and Poisson's equation in two‐dimensional domains with polygonal boundary.

Addresses the computational aspects involved in the numerical simulation of sheet stamping processes. Focuses on some numerical aspects of the intrinsic complexity of these problems, the first of which is the necessity to take into account properly membrane and bending effects. Presents a well‐adapted shell element. The second aspect concerns the description and the implementation of the initial orthotropic plastic behaviour for sheet metal parts, based on a formulation in a rotating frame using the initial microstructure rotation. The stress calculation algorithm is based on a particular implementation of the elastic predictor‐plastic corrector method. The last aspect concerns the solution procedures with a particular development concerning the treatment of the blankholder load as a constraint. A set of computational results validated with experiments prove the accuracy of the proposed approach in solving stamping problems.

Analyses several algorithms for the integration of the Jaumann stress rate. Places emphasis on accuracy and stability of standard algorithms available in commercial and government finite element codes in addition to several other proposals available in the literature. The analysis is primarily concerned with spinning bodies and reveals that a commonly used algorithm is unconditionally unstable and only first‐order objective in the presence of rotations. Other proposals are shown to have better accuracy and stability properties. Finally, shows by example that even a consistent and unconditionally stable integration of hypoelastic constitution does not necessarily yield globally stable finite element simulations.

To propose novel predictor‐corrector time‐integration algorithms for pseudo‐dynamic testing.

The novel predictor‐corrector time‐integration algorithms are based on both the implicit and the explicit version of the generalized‐α method. In the non‐linear unforced case second‐order accuracy, stability in energy, energy decay in the high‐frequency range as well as asymptotic annihilation are distinctive properties of the generalized‐α scheme; while in the non‐linear forced case they are the limited error near the resonance in terms of frequency location and intensity of the resonant peak. The implicit generalized‐α algorithm has been implemented in a predictor‐one corrector form giving rise to the implicit IPC‐ρ_∞ method, able to avoid iterative corrections which are expensive from an experimental standpoint and load oscillations of numerical origin. Moreover, the scheme embodies a secant stiffness formula able to approximate closely the actual stiffness of a structure. Also an explicit algorithm has been implemented, the EPC‐ρ_b method, endowed with user‐controlled dissipation properties. The resulting schemes have been tested experimentally both on a two‐ and on a six‐degrees‐of‐freedom system, exploiting substructuring techniques.

The analytical findings and the tests have indicated that the proposed numerical strategies enhance the performance of the pseudo‐dynamic test (PDT) method even in an environment characterized by considerable experimental errors. Moreover, the schemes have been tested numerically on strongly non‐linear multiple‐degrees‐of‐freedom systems reproduced with the Bouc‐Wen hysteretic model, showing that the proposed algorithms reap the benefits of the parent generalized‐α methods.

Further developments envisaged for this study are the application of the IPC‐ρ_∞ method and of EPC‐ρ_b scheme to partitioned procedures for high‐speed pseudo‐dynamic testing with substructuring.

The implicit IPC‐ρ_∞ and the explicit EPC‐ρ_b methods allow a user to have defined dissipation which reduces the effects of experimental error in the PDT without needing onerous iterations.

The paper proposes novel time‐integration algorithms for pseudo‐dynamic testing. Thanks to a predictor‐corrector form of the generalized‐α method, the proposed schemes maintain a high computational efficiency and accuracy.

The results from a direct numerical simulation (DNS) of turbulent, incompressible flow through a square duct, with an imposed temperature difference between the horizontal walls, are presented. The vertical walls are assumed perfectly insulated, and the Reynolds number, based on the bulk velocity and the hydraulic diameter, is about 4400. Our results indicate that secondary motions do not affect dramatically global parameters, like the friction factor and the Nusselt number, with respect to the plane‐channel flow, but the distributions of the local shear stress and heat flux at the walls are highly non‐uniform, due to the presence of these secondary motions. It is also shown that an eddy‐diffusivity approach is capable to reproduce well the turbulent heat flux. All simulations were performed by an efficient finite volume algorithm. A description of the numerical algorithm, together with an analysis of time‐accuracy, is included. The OpenMP parallel programming language was exploited to obtain a moderately‐scalable application.