by Pedro V. Marcal
The writer was previously engaged in the development of a general purpose program for Structural Mechanics.. The general framework defined then for a general purpose program served as a blueprint for the development of the MARC program and in turn concepts of general purpose programs were extended and clarified by the developments in the program.
In this discussion we consider the effects of introducing a fluid finite element based on the Eulerian frame of reference and its insertion into what was previously a Lagrangian frame of reference world. The discussion will be given specificity by considering how the concept applies to the Multi-Physics Adaptive Computer Technology (Mpact) general purpose program.
In MPACT the fluid element is based on the optimal Least Squares Finite Element Analysis(LSFEA)  method. Each node has 7 degrees of freedom , namely 3 velocities vi , 3 vorticities wi and a pressure degree of freedom. The solution of the fluid flow equations are based on the solution of the total degrees of freedom of v,w and p. On the solids side we have only the three displacements ui.
The formulation on the Lagrangian solids is made in terms of nonlinear incremental analysis and is cast in terms of an increment in time so that the degrees of freedom solved for are effectively the velocity, D0uk. Where the subscript D0 denotes differentiation in time in the Lagrangian frame. The velocity of a moving particle in the fluid domain is given by vdjuk+d0uk. Where dj denotes partial differentiation w.r.t. the coordinate directions and d0 denotes partial differentiation w.r.t. time in the Eulerian Frame.
A fluid particle that is now at the fluid solid interface (ie a common node) may be thought of as having a zero value of v as well as djuk in the solid volume so that at the interface we have d0uk and D0uk.and since this is the same particle, we can equate the two Eulerian and Lagrangian degrees of freedom. The coupling is automatic and we will term this Automatic Eulerian Lagrangian coupling (AEL) in similar fashion to the more popular ALE methods.
Finally we note that this method allows us to couple the degrees of freedom directly in what is generally referred to as strong coupling. This allows us to better handle the highly nonlinear interactive effects between the fluid and solid interaction.
Heat Transfer occurs in both the Fluid and the Solids field. The formulations must be made in different frames of reference for the two dimains. In the Eulerian frame, the rate of change of temperature of a particle is given by the material rate of change
T = vdjT + d0T
In order to capture the temperature field in a rapidly moving fluid field we must include the djT ie the flux terms. We have found that when the conservation equations are solved without introducing the flux terms we converge to a wrong answer. This is particularly evident in the solution of natural convection problems. In fluid flow terminology the solution of the heat transfer problem is called a species or a transport problem.
In these transport solutions diffusion and other exchanges are allowed to take place in the moving velocity streams. Many technologically important problems are handled by species degrees of freedoms such as multi-phase mixing, k-epsilon formulation for turbulence and etc.
The fluid flow equatioins give rise to two forms of eigenvalue equations that parallel the eigenvalue problems in solids. We write the fluid equations in matrix form as
From this we obtain the buckling eigenvalue problem for the steady state.
assuming linear variation of the Convective Matrix C w.r.t. djvk. For more accuracy we may adopt the in situ buckling procedure where the solution is calculated near the buckling point. Note that this equation clearly tells us that there is a limit on the laminar flow. The instability evidences itself in formation of vortices which are then shed in an unsteady flow. and for the next eigenvalue problem we have the modal equation
Note that the convective matrix C plays the same role here as the geometric stiffness matrix in the equivalent formulation of the natural frequency analysis of solids.
In the general purpose concept, we describe our features as components of several libraries. It is the developer’s responsibility to see that these libraries are consistent and interface correctly with each other as well as within each library. The inclusion of multi-physics results in a Domain Library. MPACT's Domain Library is currently given by the following:
- Stress Continuum field , Lagrangian frame.
- Fluid Flow field, Eulerian frame
- Heat Transfer field, Eulerian .
- Heat Transfer, Lagrangian frame
Analog fields as subset of above
- Harmonic field (electro-static, seepage)
- Biharmonic field (electro-magnetic)
Also in the solution of the resulting nonlinear FE equations it is convenient to introduce a library of solvers where we can mix direct solutions and iterative solutions in various phases of the problem solution. In terms of the accuracy of the solution, it is important to introduce a library of adaptive control functions where limits are placed on the acceptable error norms. MPACT’s adaptive control library is given by the following.
- Adaptive geometry, meshing control
- Adaptive loading, time stepping control
- Adaptive tangent arc loading, Riks and Crisfield methods
- Adaptive field error control
Because the program can measure the respective errors at any stage of the analysis, it is incumbent on the program to protect the user from any possible causes of error such as mesh size and placement and loading increments. All the user needs specify is the level of error that can be tolerated.
The commonly used error norms for the field solutions such as those used by Zienkiewicz and Zhu are often based on local measures. It has been found useful to weigh these error norms in terms of important variables such as equivalent stress. This gives it a global character and saves refinement effort where the solution is of little interest. In the adaptive control of field errors, the writer has found that the subdivision of tetra meshes into hex meshes results in considerable improvements in accuracy.
The material behavior is the area in which the interdependence of material and state variables such as (temperatures, equivalent stress and etc.) result in explosive growth of possible combinations. In MPACT we handle this by formally introducing a multiplier function where the multiplier is a function of any of the state variables.
In a multi-physics environment, we have observed the need for at least two multiplier functions that may be combined either as a sum or a multiplication sequence.
We summarize the traditional libraries with the components that make up each of them.
- Domain Library (5 components)
- Adaptive Library (4 components)
- Function Library (11 components)
- Element Library (21 components)
- Material Library (30 components)
- BC Features Library(15 components)
- Geometric Entities Lib. from solid model (16 components)
- Solvers (4 components)
Now we are able to estimate the number of different types of problems that can be solved by a general purpose program. Reasoning as before, we may select 3 components (6 for material library because of multiplier functions) from each library and combine them in a consistent manner. We obtain.
Solution types=5C3 * 4C3 * 11C3 * 21C3 * 30 C6 * 14C3 * 14C3 * 4C2 =1.06 E 20
Where the notation mCn means a combination of m components in any of n choices.
This can be compared to the number of analysis types of 1.0 E 12 found for MARC.
The documentation and training assistance for such a feature rich program presents a large challenge. The answer to this challenge lies in a systematic organization of the GUI to present the Library choices in clear and unambiguous form.
This is enhanced by context sensitive help backed up by electronic documentation. The goal of the GUI design is to enable a user, familiar with just the outlines of the menu system, to complete an FEA model without having to read a manual. Such a goal of GUI design is achieved in programs such as Microsoft Excel and Microsoft Word. The majority of users of such programs do so without having read any detailed documentation.
Though such programs do not rival the MPACT program in technical difficulty, their feature rich options are just as challenging in GUI design. The secret to their success is that they adhere to the Windows standards and conventions for GUI design and use so that all users within the Windows OS community have become quite comfortable with their use.
The development of MPACT’s GUI has also adhered strictly to these design standards and conventions and the same benefits are obtained from its use.
- Hibbitt, H.D., Levy, N.J. and Marcal , P.V. ”On General Purpose Programs For Nonlinear Finite Element Analysis”, Proceedings of Symposium on General Purpose Finite Element Computer Programs, American Society of Mechanical Engineers Programs, Edited by Marcal, P. V. , November 1970.
- Jiang, B.N. “The Least-Squares Finite Element Method”,Springer-Verlag,Berlin,1998.
- Zienkiewicz, O.C. and Zhu, J.Z. “A Simple Error Estimator and Adaptive Procedure for practical Engineering Analysis” , Int. J. Num. Meth. Eng.,24,335-57 , 1987.