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

https://www.uni-due.de/~hm0014/ag_neff/publikationen/ijss_hartmann_neff03.pdf

Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility

For the numerical solution of materially non-linear problems like in computational plasticity or viscoplasticity the finite element discretization in space is usually coupled with point-wise defined evolution equations characterizing the material behaviour. The interpretation of such systems as differential–algebraic equations (DAE) allows modern-day integration algorithms from Numerical Mathematics to be efficiently applied. Especially, the application of diagonally implicit Runge–Kutta methods (DIRK) together with a Multilevel-Newton method preserves the algorithmic structure of current finite element implementations which are based on the principle of virtual displacements and on backward Euler schemes for the local time integration. Moreover, the notion of the consistent tangent operator becomes more obvious in this context. The quadratical order of convergence of the Multilevel-Newton algorithm is usually validated by numerical studies. However, an analytical proof of this second order convergence has already been given by authors in the field of non-linear electrical networks. We show that this proof can be applied in the current context based on the DAE interpretation mentioned above. We finally compare the proposed procedure to several well-known stress algorithms and show that the inclusion of a step-size control based on local error estimations merely requires a small extra time-investment. Copyright © 2001 John Wiley & Sons, Ltd.

Remarks on the interpretation of current non‐linear finite element analyses as differential–algebraic equations

This paper deals with two main topics. The first one concerns the equivalence of stress algorithms, based on a Backward-Euler-step applied on viscoplastic models of Chaboche-type, and their elastoplastic counterpart. Generally, the stress algorithm yields a system of non-linear algebraic equations and the corresponding consistent tangent operator, occurring in the principle of virtual displacements, leads to a system of linear equations. This procedure can be obtained utilizing only numerical methods. The second topic concerns a special constitutive relation based on a kinematic hardening model using a sum of Armstrong/Frederick terms, which is equivalent to a multi-surface plasticity model. Applying this model a so-called problem-adapted stress algorithm is derived, where only one non-linear equation must be solved. This result is independent of the number of terms in the hardening model. Furthermore, only the viscoplastic algorithm must be implemented, since it includes the elastoplastic constitutive model as a special case. © 1997 by John Wiley & Sons, Ltd.

An efficient stress algorithm with applications in viscoplasticity and plasticity

Usually the notion “Newton-Raphson method” is used in the context of non-linear finite element analysis based on quasi-static problems in solid mechanics. It is pointed out that this is only true in the case of non-linear elasticity. In the case of constitutive equations of evolutionary-type, like in viscoelasticity, viscoplasticity or elastoplasticity, the “Multilevel-Newton algorithm” is usually applied yielding the notions of global and local level (iteration), as well as the consistent tangent operator. In this paper, we investigate the effects of a consistent application of the classical Newton-Raphson method in connection with the finite element method, and compare it with the classical Multilevel-Newton algorithm. Furthermore, an improved version of the Multilevel-Newton method is applied.

Stefan Hartmann

Papers

Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility

Remarks on the interpretation of current non‐linear finite element analyses as differential–algebraic equations

An efficient stress algorithm with applications in viscoplasticity and plasticity

A remark on the application of the Newton-Raphson method in non-linear finite element analysis

Computation in finite-strain viscoelasticity: finite elements based on the interpretation as differential–algebraic equations