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

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)

Pile Foundations for Offshore Wind Turbines. Numerical and Experimental Investigations.

The thermal coupling of a fluid and a structure is of great significance for many industrial processes. As a model for cooling processes in heat treatment of steel we consider the surface coupling of the compressible Navier-Stokes equations bordering at one part of the surface with the heat equation in a solid region. The semi-discrete coupled system is solved using stiffly stable SDIRK methods of higher order, where on each stage a fluid-structure-coupling problem is solved. For the resulting method it is shown by numerical experiments that a second order convergence rate is obtained. This property is further used to implement a simple time-step control, which saves considerable computational time and, at the same time, guarantees a specified maximum error of time integration.

A time-adaptive fluid-structure interaction method for thermal coupling

Within the last decade, very sophisticated numerical methods for the iterative and partitioned solution of fluid–structure interaction problems have been developed that allow for high accuracy and very complex scenarios. The combination of these two aspects–accuracy and complexity–demands very high computational grid resolutions and, thus, high performance computing methods designed for massively parallel hardware architectures. For those architectures, currently used coupling methods, which mainly work with a staggered execution of the fluid and the structure solver, i.e., the execution of one solver after the other in every outer iteration, lead to severe load imbalances: if the flow solver, e.g., scales on a very large number of processors but the structural solver does not due to its limited amount of data and required operations, almost all processors assigned to the coupled simulations are idle during the execution of the structure solver. We propose two new iterative coupling methods that allow for the simultaneous execution of flow and structure solvers. In both cases, we show that pure fixed-point iterations based on the parallel execution of the solvers do not lead to good results, but the combination of parallel solver execution and so-called quasi-Newton methods yields very efficient and robust methods. Those methods are known to be very efficient also for the stabilization of critical scenarios solved with the standard staggered solver execution. We demonstrate the competitive convergence of our methods for various established benchmark scenarios. Both methods are perfectly suited for use with black-box solvers because the quasi-Newton approach uses solely input and output information of the solvers to approximate the effect of the unknown Jacobians that would be required in a standard Newton solver.

Parallel coupling numerics for partitioned fluid–structure interaction simulations

On plastic incompressibility within time-adaptive finite elements combined with projection techniques

An accurate prediction of the temperature distribution in space and time plays an important role in many industrial applications, in particular when phase transformations are involved. In this article the thermo-physical properties of steel 51CrV4 (SAE 6150) are determined and used in numerical simulations. For the simulation of the temperature field a semi-discrete approach is used, consisting of a finite element approximation in space and a high order Runge--- Kutta integration in time. Several adaptive high-order time integration method (stiffly accurate diagonally implicit Runge---Kutta methods) are applied and their computational efficiency is investigated. The theoretical rates of convergence are achieved for all problems, including the non-linear case. Whereas the second order accurate method of Ellsiepen with time adaptive step-size control proves to be most efficient. Further, the influence of the material model on the simulation results is studied and the numerical results are verified by experiments. The best correlation of the simulation and experimental data is achieved using temperature-dependent parameters.

Experimental validation of high-order time integration for non-linear heat transfer problems

On the one hand, displacement controlled processes are frequently applied in Computational Mechanics using the finite-element method, on the other hand, there are some shortcomings regarding the study and the presentation of this particular case. In this article, we focus our theoretical considerations on quasi-static problems with constitutive equations of evolutionary type. In this case, it is currently known that the solution procedure of global and local iterations within the “Newton-Raphson method” is related to the method of lines, where one arrives at a system of differential-algebraic equations after the spatial discretization by means of the finite element method. This could be solved by means of the Backward-Euler method or more appropriately using time-adaptive, stiffly accurate, diagonally implicit Runge-Kutta methods in combination with the Multilevel-Newton method. In this respect the theoretical basis of currently applied implicit non-linear finite element analyses is extended to displacement controlled processes. In this article, the calculation of the reaction forces using the method of Lagrange multipliers and the penalty method are of special interest in view of the new DAE-approach. Accordingly, an important objective lies in a consistent notation so that the dependence of known and unknown variables as well as the transition from local to global level becomes obvious. These investigations are of utmost importance for micro-macro homogenization techniques in FE$^2$ approaches and the development of new global time and space-adaptive integration methods.

Displacement control in time-adaptive non-linear finite-element analysis

The method of vertical lines in the case of quasi-static solid mechanics applying constitutive models of evolutionary-type yields after the spatial discretization by means of finite elements a system of differential-algebraic equations. It is of substantial interest how fast, accurate, and stable such computations can be carried out. Moreover, the questions are how simple the implementation can be done and how susceptible a procedure is to programming errors. In this article, this is investigated for half-explicit Runge–Kutta methods that are applied to small and finite strain viscoelasticity. The advantage of the method is given by a non-iterative scheme on element level. Additionally, it turns out that for models where linear elasticity is one ingredient in the constitutive model, the method leads to only one required LU-decomposition at the beginning of the entire computation, and in each time step, only one back-substitution step has to be carried out. This outperforms current finite element computations. Order investigations of various integration schemes and the automatic step-size behavior are studied. This new proposal is compared to a classical Backward-Euler implementation, high-order stiffly accurate diagonally implicit Runge–Kutta, and recently proposed Rosenbrock-type methods. Advantages and disadvantages of the applied schemes are discussed.

Karsten J. Quint

Papers

A time-adaptive fluid-structure interaction method for thermal coupling

On plastic incompressibility within time-adaptive finite elements combined with projection techniques

Experimental validation of high-order time integration for non-linear heat transfer problems

Displacement control in time-adaptive non-linear finite-element analysis

Comparison of diagonal-implicit, linear-implicit and half-explicit Runge–Kutta methods in non-linear finite element analyses