Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

During the last decade, a big progress has been achieved in the analysis and numerical treatment of Initial Value Problems (IVPs) in Differential Algebraic Equations (DAEs) and Ordinary Differential Equations (ODEs). In spite of the rich variety of results available in the literature, there are still many specific problems that require special attention. Two of such, which are considered in this work, are the optimization of order of accuracy and reduction of cost of functional evaluations of Explicit Runge - Kutta (ERK) methods.

Traditionally, the maximum attainable order p of an s-stage ERK method for advancing the solution of an IVP is such that
p(s)
 4
 
In 1999, Goeken presented an s-stage ERK Method of order p(s)=s +1,s>2.
However, this work focuses on the design of a new approximation algorithm that reduces the cost of functional evaluations and yet increases the attainable order higher 
U n and Jonhson [94]; and the classical ERK methods. The order p of 
the new scheme called Multiderivative Explicit Runge-Kutta (MERK) Methods is such that p(s) 2. The stability, convergence and implementation for the optimization of IVPs in DAEs and ODEs systems are also considered.

Numerical solution of initial value problems in differential - algebraic equations

A flux splitting scheme is proposed for the general nonequilibrium flow equations with an aim at removing numerical dissipation of Van-Leer-type flux-vector splittings on a contact discontinuity. The scheme obtained is also recognized as an improved Advection Upwind Splitting Method (AUSM) where a slight numerical overshoot immediately behind the shock is eliminated. The proposed scheme has favorable properties: high-resolution for contact discontinuities; conservation of enthalpy for steady flows; numerical efficiency; applicability to chemically reacting flows. In fact, for a single contact discontinuity, even if it is moving, this scheme gives the numerical flux of the exact solution of the Riemann problem. Various numerical experiments including that of a thermo-chemical nonequilibrium flow were performed, which indicate no oscillation and robustness of the scheme for shock/expansion waves. A cure for carbuncle phenomenon is discussed as well.

A Flux Splitting Scheme with High-Resolution and Robustness for Discontinuities(Proceedings of the 12th NAL Symposium on Aircraft Computational Aerodynamics)

The Physics of Rubber Elasticity (J. Meixner)

https://hal.archives-ouvertes.fr/hal-00457010/document

B¯ and F¯ projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order NURBS elements

The derivation of Rosenbrock-Krylov methods for index 1 DAEs involves two well known techniques: a limit process which transforms a singular perturbed ODE to an index 1 DAE and the use of Krylov iterations instead of direct linear solvers for the stage equations. We show that our derived class of Rosenbrock-Krylov schemes is independent of the order in which we apply these techniques. We also conclude that for convergence a rather accurate solution of the algebraic part is always needed.

/pdf/time-integration-of-index-1-daes-with-rosenbrock-methods-20mnsc0qso.pdf

Time integration of index 1 DAEs with Rosenbrock methods using Krylovsubspace techniques

Composite materials are widely used in different industrial fields, because of their good formability and their high strength to weight ratio. In the present work a triple-layered sandwich composite is investigated. Experimental tests at room temperature are carried out for the materials constituting the composite. A finite element model of a deep-drawing process of the composite is performed, where a finite strain constitutive model for the metal part, with material parameters calibrated to uniaxial tensile tests, has been implemented. Experimental results are compared to the numerical simulations in view of validation purposes. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

Deep drawing of a layered composite: material characterization and finite‐element simulation

The consistent application of the time-space discretization in the case of quasi-static structural problems based on constitutive equations of evolutionary type yields after the finite element discretization to a system of differential-algebraic-equations (DAE). In order to carry out the time discretization, time-adaptive Rosenbrock-type methods are applied to the DAE-system, which offer the possibility of a completely iteration-less procedure. This presentation shows the behavior of a new global finite element approach and compares it to the classical implicit (iterative) procedure. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Application of Rosenbrock-type methods to a constitutive model of polymeric material

In this paper we interpret the finite-element method applied to constitutive models with internal variables as the solution of differential algebraic equations (DAE). With this interpretation we can utilize Diagonal Implicit Runge-Kutta methods (DIRK) as well as a corresponding step-size control and a special solution technique for block-structured systems of equations. This proceeding doesn't affect already developed FE-implementations which are based on the Backward Euler method. Furthermore, the meaning of the consistent tangent operator becomes more obvious. The paper ends with some examples in creep and relaxation tests, where step size control should always be used conditioned by the different time scales.

A time adaptiv finite-element procedure applied to creep and relaxation processes

https://onlinelibrary.wiley.com/doi/pdf/10.1002/pamm.201610186

Stefan Hartmann

Papers

Time integration of index 1 DAEs with Rosenbrock methods using Krylovsubspace techniques

Deep drawing of a layered composite: material characterization and finite‐element simulation

Application of Rosenbrock-type methods to a constitutive model of polymeric material

A time adaptiv finite-element procedure applied to creep and relaxation processes

Experimental and Numerical Study of Polypropylene/Polyethylene Thin Foil in Biaxial Tension