Copyright © 2004 Shevchenko

The purpose of these pages is to make information available to the general academic and commercial community, related to meshless technology, R-function approach, 3D solvers and geometric engines, numerical computation, computer graphics, mathematical modeling, computer simulation, and engineering analysis.

Please feel welcome to E-mail me with questions, suggestions, or feedback.

What is FIELD?

Software Highlight

At Least Five Benefits from FIELD System:

Background

Effective Application of the FIELD System and R-functions Method

Geometry Engine

Optimization

Temperature Fields (Thermal Analysis)

Deformation and Force Fields (Stress-Strain, Frequency and Buckling Analysis)

Tire Simulation

Computation Nanotechnology

Download

Links

FIELD system is a medium for interactive modeling and simulation of physical and mechanical fields (temperature, deformation, force, electro-magnetic, etc.). FIELD is a comprehensive simulation system that provides accurate, reliable, and ease of use for engineers and scientists. FIELD is a cross platform mathematical C++ library which supports new meshlees technology from the R-functions theory, advanced numerical computing, 3D graphical plotting and animation, etc. Moreover, a FIELD has many salient features: FIELD can be run directly into any CAD environment without difficult meshing process, interactive solution of different complicated optimization problems with respect to geometrical parameters, etc.

FIELD is a high performance 3D solver that enables engineers to quickly and effectively create meshless mathematical models for engineering simulations and analysis

FIELD represents the true state-of-the-art in R-functions analysis and 3D solid modeling. FIELD is designed to integrate easily with tomorrow’s technologies.

- Speed & Accuracy
- Using OpenGL
- Support of CSG
- Meshless
- Optimization

- Extremely complicated models can be analyzed 10-100 times faster than finite elements solvers do.
- FIELD uses both boundary representation and constructive solid geometry schemes for the analysis model and requires absolutely no finite element, grid generation, nor meshing.
- FIELD creates parameterized solution structures satisfy all boundary condition exactly and provides computation support for problems with changing geometry and different physical parameters.
- FIELD allows for use of special solutions functions for inexpensive and very accurate resolution of specific model features (cracks, holes, spot welds, etc.).
- FIELD can be used in conjunction with most basic function and numerical procedures, including other meshfree methods and can be run into modern CAD/CAM systems for the solution various engineering problems.

The theory of R-functions (Rvachev's functions) describes geometric objects and domains by an m -times continuously differentiable real functions that determine the boundary, the interior, and other properties of the modeled object. In other words, due to R-functions we know how to write the equations of complicated geometric objects, to create generalization of classical Lagrange-Teylor-Hermite formulae, and to construct the solution structures for several boundary value problems and solve them without meshing.

The theory of R-functions, its application to numerous engineering problems , the methodology, and computer software have been developed in Ukraine over last thirty years. This investigation is described in numerous books and article, including some fundamental introduction to R-functions published in English and some new papers which explore and explain how to use R-functions for solid modeling in modern CAD/CAM systems.

FIELD system is the computer implementation of R-functions theory which includes software for transformation geometric, analytic and graphical information, automatic differentiation, computational integration and solving of linear algebra problems. FIELD system is intended for scientists and engineers who need to solve mathematical modeling and simulation problems.

- Solution of partial differential equation in unbounded domain.

- Shape and topology optimization.

- Solution of some inverse problems.

- Investigation for propagate of cracks.

- Solution some typical concentration problem.

- Simulation of deforming domains (time-varying geometry and boundary conditions).

Constructive Solid Geometry (CSG) is an approach to geometric modeling which applies Boolean operations
on the set of solid primitives. In a recent development, work by Nigel Stewart has shown that Sequenced
Convex Subtraction (SCS) can be used to perform fast rendering of intersected geometric objects.
It has been implemented using the Z and stencil testing functionality of OpenGL. In this example we use simple
subtraction set of spheres from cube. (Click on the picture for details).

FIELD system allows to create the equations of complicated objects. In other words, we can automatically
construct the function, positive inside the object, equal to zero on the boundary, and negative outside the object.
FIELD system can create also the equations of segments. In this case we can construct the function equal to zero on the
boundary and positive everywhere else. On these pictures we can see such functions with different differential properties.
(Click on the picture for details).

FIELD system can use not only CSG rendering but can also use a simple geometric engine with R-functions to
implicitly represent of complicated objects. In this example we constructed the function with the following
properties: it is positive inside of the object, equal to zero on the boundary, and negative outside of the
object. Therefore, FIELD creates function Omega = Sphere - Cylinder1 - Cylinder2 and uses it to solve
boundary value problems. (Click on the picture for details).

In FIELD system we can not only create the equation of polygon but also construct some function with desire properties.
In other words, we can do generalize interpolation inside the polygon. We can construct the function
which equal to given functions on the boundary. Here we can see some pictures for illustration some
extrude - revolve - helical solid modeling possibilities. So, FIELD can not only create different complicated
curves, surfaces and bodies, but also construct some functions which everywhere defined and satisfy
the given boundary conditions. (Click on the picture for details).

In the FIELD system we can solve a large class of so-called extreme value problems: we desire to find
parameter values or functions which minimize or maximize a quantity dependent upon them. In this example we
constructed function, which equal to zero on the boundary of concave polygon. On the next step we added new
primitives (circles) to our solution. And finally, we decreased maximal value of function in the domain.
(Click on the picture for details).

Here is the example for optimization in 3-dimensional space. A function is created that satisfies
boundary conditions: maximum value on the two intersection cylinders. These cylinders can then
be translated and rotated and the function’s value can then be calculated at each step of the
simulation. Furthermore, FIELD can solve problems with different geometric parameters and boundary
conditions choosing the best solution for our needs. (Click on the picture for details).

Two-dimensional and three-dimensional stationary, non-stationary, linear, non-linear problems can be solved.
The following figures illustrates some advantages of meshless technology.

The coordinates of the hole center in the domain is the main difference between these objects.
For the solution such problems using standard finite elements solvers we have to create new mesh for each
objects and apply new boundary conditions. You can see such mesh and temperature distribution on the pictures.
(Click on the picture for details).

From the R-functions point of view we have no difference between geometry. Due to R-functions we do not
need to calculate new mesh for such regions and apply new boundary conditions.
We need only to change coordinates of cylinder’s centers and calculate new results.
Moreover, we can do it automatically for different geometrical and physical
parameters and create animations with respect to geometric changing. (Click on the picture for details).

A lot of practical engineering situations require solving problems with time varying geometry and
boundary conditions. A 3-dimensional example of such problem is illustrated in this demo. Modeling
and simulation the space and time temperature distribution for this problem require solving a non-stationary
heat transfer problem. (Click on the picture for details).

Problems involving the solidification or melting of metals are of considerable importance in
casting processes. In FIELD system we can solve 3-dimensional, non-linear, non-stationary boundary
value problems with the change in the phase. In this example we can see temperature distribution for
different time (transient animation). (Click on the picture for details).

The FIELD system was used for solving problem of thin plate bending and computation of plate’s natural frequencies
and natural vibration modes. It is described by the bi-harmonic equation with mixed boundary conditions.
The set of eigen values is a frequency spectrum, every eigen function defines the oscillation.
(Click on the picture for details).

In this example we solved problem about how the honeycomb is used as an energy absorber under impact
loads. FIELD system allows us to create the equation of honeycomb (mathematical model), applies all
boundary conditions and calculates deformation, stresses and other characteristic. The deformed shapes
for the impact simulation are shown in this animation. (Click on the picture for details).

We developed R-functions tire models for prescribed displacements over irregular terrain.
In this case we can not only take into account complicated geometric information, but also use some
experimental data about dynamic behavior. Moreover, we can dynamically create solution for exact
modeling of interaction between wheels and surfaces of motion. (Click on the picture for details).

There are a lot of different physical, mechanical, chemical processes which we can describe
by boundary value problems with partial differential equations.
For example, if we need to simulate well-known Brownian motion,
we can use some mathematical models with so-called stochastic differential equation.
We can do modeling and simulation in nanomechanics, exploring such properties as Yong's
modulus, bending stiffness, buckling criteria, and tensile and compressive
strengths, etc. (Click on the picture for details).

- FieldDemo.zip (1449 K) Binary
- Field.zip (912 K) Source

R-functions and Applications

- R-functions - Academician Vladimir Logvinovich Rvachev
- MeshFree Analysis - Spatial Automation Laboratory

Methods, Algorithms, Programs and Tools

- Web-Spline - Prof. Dr. Klaus Hollig
- CSG - Dr. Nigel Stewart
- Microsoft - Microsoft Visual Studio .NET 2003
- Fraps - Real-Time Video Capture