COP: a data library of optical properties of hexagonal ice crystals

The data library of optical properties of hexagonal ice crystals for radiative modeling, Cirrus Optical Properties (COP), is introduced. It includes phase functions, asymmetry parameters, extinction cross sections, and single scattering albedos. Furthermore, lidar ratios and depolarization are given. The dependence of these parameters on wavelength, particle size, and shape is calculated, and different particle orientations are considered. In addition, a simple FORTRAN code is provided to calculate the corresponding properties of size distributions. Thus the data library is a very flexible tool for determining the optical parameters of ice clouds for climatological purposes and remote sensing. The data library and the FORTRAN code are distributed through electronic mail.     

Explicit dynamic analysis of elasto-plastic laminated composite shells: implementation of non-iterative stress update schemes for the HOFFMAN yield criterion

The dynamic analysis of composite shell structures is carried out by an explicit finite element code employing 4-node one-point quadrature elements. The anisotropic Hoffman yield criterion is adopted to model the laminates. The formulation for stress update using a backward Euler scheme is presented in the plane stress subspace. Several numerical examples are presented. The issue of implementing single-iteration schemes for stress update is also investigated.     

Numerical implementation of constitutive integration for rate-independent elastoplasticity

In this paper, constitutive integration for rate-independent, small deformation elastoplasticity is studied. Smooth yield surfaces and work/strain hardening are assumed. Both associative or non-associative flow rules are considered. An Euler backward algorithm is applied for constitutive integration. Tangent moduli that are consistent with the Euler backward algorithm, i.e. a so-called consistent tangent operator, are derived. Emphasis is placed on numerical implementation of the Eular backward algorithm into finite element codes using such a consistent tangent operator. In particular, a commercial code ANSYS is considered. Numerical examples, including materials sensitive and insensitive to hydrostatic stress, are used for the verification of the implementation. A comparison of the algorithmic performance to an explicit Euler forward algorithm is given and the superiority of the Euler backward algorithm is demonstrated.The work described in the present paper has been sponsored by The Research Council of Norway, The North Calotte Education and Research Council, Statoil and Norsk Hydro.     

Numerical implementation of anisotropic damage mechanics

This paper describes implementation of anisotropic damage mechanics in the material point method. The approach was based on previously proposed, fourth‐rank anisotropic damage tenors. For implementation, it was convenient to recast the stress update using a new damage strain partitioning tensor. This new tensor simplifies numerical implementation (a detailed algorithm is provided) and clarifies the connection between cracking strain and an implied physical crack with crack opening displacements. By using 2 softening laws and 3 damage parameters corresponding to 1 normal and 2 shear cracking strains, damage evolution can be directly connected to mixed tensile and shear fracture mechanics. Several examples illustrate interesting properties of robust anisotropic damage mechanics such as modeling of necking, multiple cracking in coatings, and compression failure. Direct comparisons between explicit crack modeling and damage mechanics in the same material point method code show that damage mechanics can quantitatively reproduce many features of explicit crack modeling. A caveat is that strengths and energies assigned to damage mechanics materials must be changed from measured material properties to apparent properties before damage mechanics can agree with fracture mechanics.     

Higher-order sensitivity analysis of finite element method by automatic differentiation

To design optimal mechanical structures, design sensitivity analysis technique using higher order derivatives are important. However, usual techniques for computing the derivatives, for example numerical differentiation methods, are hard to apply to real scale structures because of the large amount of computational time and the accumulation of computational errors.To overcome the problem, we have studied a new approach for higher order sensitivity analysis of the finite element method using automatic differentiation techniques. The method automatically transforms FORTRAN code to special purpose code which computes both the value of the given functional dependence and their derivatives. The algorithm used in the method automatically and efficiently computes accurate values of higher order partial derivatives of a given functional dependence on many variables.This paper reports the basic principles of the automatic differentiation method and some experiments on the sensitivity analysis of mechanical structures. The original program of structural analysis by the finite element method is implemented in FORTRAN, which is developed by the first author. Using the proposed method, we get more accurate sensitivity and prediction values compared with usual numerical differentiation. We also discuss the effectiveness of the proposed approach for the sensitivity analysis of the mechanical structures.     

Computational implementation of a novel constitutive model for multidirectional composites

This paper details the numerical implementation of a constitutive model for unidirectional (UD) polymer-matrix fibre-reinforced composites, which is able to accurately represent the full non-linear mechanical response. Features such as hydrostatic pressure sensitivity, the effect of multiaxial loading and the dependence of the yield stress on the applied pressure are often neglected in constitutive modelling, but are included in this model.The constitutive model includes a novel yield function, non-associative flow rule and a non-linear kinematic hardening rule. It is combined with suitable failure criteria and associated damage model. The complete model is implemented in an explicit finite element code.Experimental test data is used to show that the model is able to predict the non-linear response of both unidirectional and multidirectional composite laminates. The model is shown to accurately predict the constitutive response under complex multiaxial loading and unloading, including significant hydrostatic pressure. Predictions are also shown to compare favourably for the evolution of matrix cracking after initial matrix cracking is detected by the failure criteria.     

Stint/CD: A stand-alone explicit time integration package for structural dynamics analysis

This paper is a user's guide for the stand-alone explicit direct time integration package STINT/CD for structural dynamics analysis. STINT/CD uses an automatic variable time increment central difference method. The purpose, function, limitations and usage of the package are described. A FORTRAN listing of STINT/CD is given along with a sample problem which illustrates its usage and performance.     

Finite element implementation of non‐linear elastoplastic constitutive laws using local and global explicit algorithms with automatic error control

Implicit stress integration algorithms have been demonstrated to provide a robust formulation for finite element analyses in computational mechanics, but are difficult and impractical to apply to increasingly complex non‐linear constitutive laws. This paper discusses the performance of fully explicit local and global algorithms with automatic error control used to integrate general non‐linear constitutive laws into a non‐linear finite element computer code. The local explicit stress integration procedure falls under the category of return mapping algorithm with standard operator split and does not require the determination of initial yield or the use of any form of stress adjustment to prevent drift from the yield surface. The global equations are solved using an explicit load stepping with automatic error control algorithm in which the convergence criterion is used to compute automatically the coarse load increment size. The proposed numerical procedure is illustrated here through the implementation of a set of elastoplastic constitutive relations including isotropic and kinematic hardening as well as small strain hysteretic non‐linearity. A series of numerical simulations confirm the robustness, accuracy and efficiency of the algorithms at the local and global level. Published in 2001 by John Wiley & Sons, Ltd.     

Computational procedures for the maximum likelihood parameters of the gamma and beta distributions

A computational procedure is presented for calculating the maximum likelihood sampled parameters for the gamma and beta distributions. The procedure employed is both computationally efficient and easy to use. In order to facilitate immediate application, a self-contained FORTRAN IV computer code has been included to carry out the necessary operations.     

Treating voxel geometries in radiation protection dosimetry with a patched version of the Monte Carlo codes MCNP and MCNPX

The question of Monte Carlo simulation of radiation transport in voxel geometries is addressed. Patched versions of the MCNP and MCNPX codes are developed aimed at transporting radiation both in the standard geometry mode and in the voxel geometry treatment. The patched code reads an unformatted FORTRAN file derived from DICOM format data and uses special subroutines to handle voxel-to-voxel radiation transport. The various phases of the development of the methodology are discussed together with the new input options. Examples are given of employment of the code in internal and external dosimetry and comparisons with results from other groups are reported.     

Combined implicit/explicit algorithms for crashworthiness analysis

In order to simulate an industrial process, an explicit method, which is conditionally stable, is the most adapted while the non-linearities evolve rapidly (impact phase, stamping process, etc.). But when the dynamics becomes quasi-linear (post-impact analysis, springback simulation, etc.), an implicit method, which is iterative, presents the advantage of unconditional stability. The optimal solution is then to have both implicit and explicit methods readily available in the same code and to be able to switch automatically from one to the other. Criteria that decide to switch from one method to another, depending on the current dynamics, have been developed. Implicit restarting conditions are also proposed that annihilate numerical oscillations resulting from an explicit calculation.     

Thermo-hydraulic analysis of the gradual cool-down to 80 K of the ITER toroidal field coil

A time-dependent thermo-hydraulic simulation for an ITER toroidal field (TF) coil gradual cool-down to 80 K has been performed using a new FORTRAN code. The code is based on a 1D helium flow and 1D multi-region solid heat conduction model. The whole TF coil is simulated taking into account thermal conduction between winding pack and case, which are cooled down separately. To limit coil mechanical stresses and coolant pressure drop in the cooling channels, an improved cool-down mode has been developed based on the analysis. Typical and gradual cool-down temperature distributions of TF coil and case are presented. The results indicate that gradual cool-down to 80 K can be achieved in 3 weeks.     

Sandwich shell finite element for dynamic explicit analysis

The contribution of this paper consists of new development of transverse shear stresses through the thickness and finding an expression for the critical time step for explicit time integration of layered shells. This work presents the finite element (FE) formulation and implementation of a higher‐order shear deformable shell element for dynamic explicit analysis of composite and sandwich shells. The formulation is developed using a displacement‐based third‐order shear deformation shell theory. Using the differential equilibrium equations and the interlayer requirements, special treatment is developed for the transverse shear, resulting in a continuous, piecewise quartic distribution of the transverse shear stresses through the shell thickness. Expressions are developed for the critical time step of the explicit time integration for orthotropic homogeneous and layered shells based on the developed third‐order formulation. To assess the performance of the present shell element, it is implemented in the general non‐linear explicit dynamic FE code DYNA3D. Several problems are solved and results are presented and compared to other theoretical and numerical results. Copyright © 2002 John Wiley & Sons, Ltd.     

Improvement of Mishchenko's T-matrix code for absorbing particles

The use of Gaussian elimination with backsubstitution for matrix inversion in scattering theories is discussed. Within the framework of the T-matrix method (the state-of-the-art code by Mishchenko is freely available at http://www.giss.nasa.gov/-crmim), it is shown that the domain of applicability of Mishchenko's FORTRAN 77 (F77) code can be substantially expanded in the direction of strongly absorbing particles where the current code fails to converge. Such an extension is especially important if the code is to be used in nanoplasmonic or nanophotonic applications involving metallic particles. At the same time, convergence can also be achieved for large nonabsorbing particles, in which case the non-Numerical Algorithms Group option of Mishchenko's code diverges. Computer F77 implementation of Mishchenko's code supplemented with Gaussian elimination with backsubstitution is freely available at http://www.wave-scattering.com.     

On the development of explicit robust schemes for implementation of a class of hyperelastic models in large-strain analysis of rubbers

The issue of developing effective and robust schemes to implement a class of the Ogden-type hypereiastic constitutive models, for large-strain analysis of rubber-like materials, is addressed. To this end, explicit forms for the corresponding material tangent-stiffness tensors are developed, and these are valid for the entire deformation range; i.e. with both distinct as well as repeated principal-stretch values. Throughout the analysis the various implications of the underlying property of separability of the strain-energy functions are exploited, thus leading to compact final forms of the tensor expressions. In particular, this facilitated the treatment of the complex cases of uncoupled volumetric/deviatoric formulations for incompressible materials, which are becoming increasingly popular in recent years. The forms derived are also amenable for use with symbolic-manipulation packages for systematic code generation.     

On the adaptive finite element analysis of the Kohn–Sham equations: methods,algorithms, and implementation

In this paper, details of an implementation of a numerical code for computing the Kohn–Sham equations are presented and discussed. A fully self‐consistent method of solving the quantum many‐body problem within the context of density functional theory using a real‐space method based on finite element discretisation of realspace is considered. Various numerical issues are explored such as (i) initial mesh motion aimed at co‐aligning ions and vertices; (ii) a priori and a posteriori optimization of the mesh based on Kelly's error estimate; (iii) the influence of the quadrature rule and variation of the polynomial degree of interpolation in the finite element discretisation on the resulting total energy. Additionally, (iv) explicit, implicit and Gaussian approaches to treat the ionic potential are compared. A quadrupole expansion is employed to provide boundary conditions for the Poisson problem. To exemplify the soundness of our method, accurate computations are performed for hydrogen, helium, lithium, carbon, oxygen, neon, the hydrogen molecule ion and the carbon‐monoxide molecule. Our methods, algorithms and implementation are shown to be stable with respect to convergence of the total energy in a parallel computational environment. Copyright © 2015 John Wiley & Sons, Ltd.     

On the effective implementation of a boundary element code on graphics processing units using an out-of-core LU algorithm

A collocation boundary element code for solving the three-dimensional Laplace equation, publicly available from http://intetec.org, has been adapted to run on an Nvidia Tesla general-purpose graphics processing unit (GPU). Global matrix assembly and LU factorization of the resulting dense matrix are performed on the GPU. Out-of-core techniques are used to solve problems larger than the available GPU memory. The code achieved about 10 times speedup in matrix assembly over a single CPU core and about 56 Gflops/s in the LU factorization using only 512 Mbytes of GPU memory. Details of the GPU implementation and comparisons with the standard sequential algorithm are included to illustrate the performance of the GPU code.     

Analysis and implementation of a new constant acceleration subcycling algorithm

An algorithm for explicit integration of structural dynamics problems with multiple time steps is proposed that averages accelerations to obtain subcycle states at a nodal interface between regions integrated with different time steps. With integer time step ratios, the resulting subcycle updates at the interface sum to give the same effect as a central difference update over a major cycle. The algorithm is shown to have good accuracy, and stability properties in linear elastic analysis similar to those of constant velocity subcycling algorithms. The implementation of a generalised form of the algorithm with non-integer time step ratios is presented. © 1997 John Wiley & Sons, Ltd.     

A FORTRAN program for profile and wavefront reduction

A FORTRAN 77 program for reducing the profile and wavefront of a sparse matrix with a symmetric structure is described. The implementation is based on an algorithm published previously by the Author and appears in response to a large number of enquiries for the source code. Extensive testing of the scheme suggests that its performance is consistently superior to that of the widely used reverse Cuthill–McKee and Gibbs–King methods. In addition to presenting a complete listing of the program, we also describe how to interface it with a typical finite element code. The scheme is especially useful in finite element analysis where it can be employed to derive efficient orderings for both profile and frontal solution schemes.