FIELD: 3D Fast Meshless Solver for Simulation and Optimization of Physical and Mechanical Fields

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.

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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
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What is FIELD?

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.

Software Highlight

At Least Five Benefits from FIELD System:

  1. Extremely complicated models can be analyzed 10-100 times faster than finite elements solvers do.
  2. FIELD uses both boundary representation and constructive solid geometry schemes for the analysis model and requires absolutely no finite element, grid generation, nor meshing.
  3. FIELD creates parameterized solution structures satisfy all boundary condition exactly and provides computation support for problems with changing geometry and different physical parameters.
  4. FIELD allows for use of special solutions functions for inexpensive and very accurate resolution of specific model features (cracks, holes, spot welds, etc.).
  5. 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.

Background

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.

Effective Application of the FIELD System and R-functions Method


Geometry Engine


animation, 280K 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).

4 pictures, 132K 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).

animation, 268K 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).

4 pictures, 134K 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).

Optimization


4 pictures, 157K 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).

two animations, 418K + 311K 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).

Temperature Fields (Thermal Analysis)

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

3 pictures, 196K 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).

animation, 469K 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).

animation, 348K 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).

animation, 233K 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).

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

FIELD system can solve all types of basic two-dimensional and three-dimensional problems (including contact ones) in theory of elasticity, thermoelastysity and plasticity for bodies of complicated configurations: homogeneous, non-homogeneous, isotropic and anisotropic ones. (Click on the picture for details).

animation, 74K 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).

animation, 359K 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).

Tire Simulation


animation, 750K 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).

Computation Nanotechnology


animation, 438K 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).

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