On the symbolic manipulation and code generation for elasto-plastic material matrices

A computerized procedure for symbolic manipulations and FORTRAN code generation of an elasto-plastic material matrix for finite element applications is presented. Special emphasis is placed on expression simplifications during intermediate derivations, optimal code generation, and interface with the main program. A systematic procedure is outlined to avoid redundant algebraic manipulations. Symbolic expressions of the derived material stiffness matrix are automatically converted to RATFOR code which is then translated into FORTRAN statements through a preprocessor. To minimize the interface problem with the main program, a template file is prepared so that the translated FORTRAN statements can be merged into the file to form a subroutine (or a submodule). Three constitutive models; namely, von Mises plasticity, Drucker-Prager model, and a concrete plasticity model, are used as illustrative examples

Lanczos eigensolution method for high-performance computers

The theory, computational analysis, and applications are presented of a Lanczos algorithm on high performance computers. The computationally intensive steps of the algorithm are identified as: the matrix factorization, the forward/backward equation solution, and the matrix vector multiples. These computational steps are optimized to exploit the vector and parallel capabilities of high performance computers. The savings in computational time from applying optimization techniques such as: variable band and sparse data storage and access, loop unrolling, use of local memory, and compiler directives are presented. Two large scale structural analysis applications are described: the buckling of a composite blade stiffened panel with a cutout, and the vibration analysis of a high speed civil transport. The sequential computational time for the panel problem executed on a CONVEX computer of 181.6 seconds was decreased to 14.1 seconds with the optimized vector algorithm. The best computational time of 23 seconds for the transport problem with 17,000 degs of freedom was on the the Cray-YMP using an average of 3.63 processors

Symbolic-numeric interface: A review

A survey of the use of a combination of symbolic and numerical calculations is presented. Symbolic calculations primarily refer to the computer processing of procedures from classical algebra, analysis, and calculus. Numerical calculations refer to both numerical mathematics research and scientific computation. This survey is intended to point out a large number of problem areas where a cooperation of symbolic and numerical methods is likely to bear many fruits. These areas include such classical operations as differentiation and integration, such diverse activities as function approximations and qualitative analysis, and such contemporary topics as finite element calculations and computation complexity. It is contended that other less obvious topics such as the fast Fourier transform, linear algebra, nonlinear analysis and error analysis would also benefit from a synergistic approach

Interval Slopes as Numerical Abstract Domain for Floating-Point Variables

The design of embedded control systems is mainly done with model-based tools such as Matlab/Simulink. Numerical simulation is the central technique of development and verification of such tools. Floating-point arithmetic, that is well-known to only provide approximated results, is omnipresent in this activity. In order to validate the behaviors of numerical simulations using abstract interpretation-based static analysis, we present, theoretically and with experiments, a new partially relational abstract domain dedicated to floating-point variables. It comes from interval expansion of non-linear functions using slopes and it is able to mimic all the behaviors of the floating-point arithmetic. Hence it is adapted to prove the absence of run-time errors or to analyze the numerical precision of embedded control systems

Material flow during the extrusion of simple and complex cross-sections using FEM

This paper deals with the extrusion of rod and shape sections and uses a 3D finite element model analysis (FEM) to predict the effect of die geometry on maximum extrusion load. A description of material flow in the container is considered in more detail for rod and shape sections in order to fully comprehend the transient conditions occurring during the process cycle. A comparison with experiments is made to assess the relative importance of some extrusion parameters in the extrusion process and to ensure that the numerical discretisation yields a realistic simulation of the process. The usefulness and the limitation of FEM are discussed when modelling complex shapes. Results are presented for velocity contours and shear stress distribution during the extrusion process. It is shown that for most of the shapes investigated, the material making up the extrudate cross-sections originates from differing regions of virgin material within the billet. The outside surface of the extrudate originates from the material moving along the dead metal zone (DMZ) and the core of the extrudate from the central deformation zone. The FE program appears to predict all the major characteristics of the flow observed macroscopically

GRACE at ONE-LOOP: Automatic calculation of 1-loop diagrams in the electroweak theory with gauge parameter independence checks

We describe the main building blocks of a generic automated package for the calculation of Feynman diagrams. These blocks include the generation and creation of a model file, the graph generation, the symbolic calculation at an intermediate level of the Dirac and tensor algebra, implementation of the loop integrals, the generation of the matrix elements or helicity amplitudes, methods for the phase space integrations and eventually the event generation. The report focuses on the fully automated systems for the calculation of physical processes based on the experience in developing GRACE-loop. As such, a detailed description of the renormalisation procedure in the Standard Model is given emphasizing the central role played by the non-linear gauge fixing conditions for the construction of such automated codes. The need for such gauges is better appreciated when it comes to devising efficient and powerful algorithms for the reduction of the tensorial structures of the loop integrals. A new technique for these reduction algorithms is described. Explicit formulae for all two-point functions in a generalised non-linear gauge are given, together with the complete set of counterterms. We also show how infrared divergences are dealt with in the system. We give a comprehensive presentation of some systematic test-runs which have been performed at the one-loop level for a wide variety of two-to-two processes to show the validity of the gauge check. These cover fermion-fermion scattering, gauge boson scattering into fermions, gauge bosons and Higgs bosons scattering processes. Comparisons with existing results on some one-loop computation in the Standard Model show excellent agreement. We also briefly recount some recent development concerning the calculation of mutli-leg one-loop corrections.Comment: 131 pages. Manuscript expanded quite substantially with the inclusion of an overview of automatic systems for the calculation of Feynman diagrams both at tree-level and one-loop. Other additions include issues of regularisation, width effects and renormalisation with unstable particles and reduction of 5- and 6-point functions. This is a preprint version, final version to appear as a Phys. Re

Material flow during the extrusion of simple and complex cross-sections using FEM

This paper deals with the extrusion of rod and shape sections and uses a 3D finite element model analysis (FEM) to predict the effect of die geometry on maximum extrusion load. A description of material flow in the container is considered in more detail for rod and shape sections in order to fully comprehend the transient conditions occurring during the process cycle. A comparison with experiments is made to assess the relative importance of some extrusion parameters in the extrusion process and to ensure that the numerical discretisation yields a realistic simulation of the process. The usefulness and the limitation of FEM are discussed when modelling complex shapes. Results are presented for velocity contours and shear stress distribution during the extrusion process. It is shown that for most of the shapes investigated, the material making up the extrudate cross-sections originates from differing regions of virgin material within the billet. The outside surface of the extrudate originates from the material moving along the dead metal zone (DMZ) and the core of the extrudate from the central deformation zone. The FE program appears to predict all the major characteristics of the flow observed macroscopically

GoSam-2.0: a tool for automated one-loop calculations within the Standard Model and beyond

We present the version 2.0 of the program package GoSam for the automated calculation of one-loop amplitudes. GoSam is devised to compute one-loop QCD and/or electroweak corrections to multi-particle processes within and beyond the Standard Model. The new code contains improvements in the generation and in the reduction of the amplitudes, performs better in computing time and numerical accuracy, and has an extended range of applicability. The extended version of the "Binoth-Les-Houches-Accord" interface to Monte Carlo programs is also implemented. We give a detailed description of installation and usage of the code, and illustrate the new features in dedicated examples.Comment: replaced by published version and reference adde