Data processing: structural design – modeling – simulation – and em – Electrical analog simulator – Of physical phenomenon
Reexamination Certificate
2000-01-24
2001-04-03
Teska, Kevin J. (Department: 2123)
Data processing: structural design, modeling, simulation, and em
Electrical analog simulator
Of physical phenomenon
C703S006000, C703S007000
Reexamination Certificate
active
06212485
ABSTRACT:
FIELD OF THE INVENTION
This invention relates to a method of analysis and optimal design for a structure made of a micro-inhomogeneous material, such as composite material, ceramics, concrete, geological material (such as rock and soil) and polycrystalline material including metal, by using a unified procedure of a molecular simulation method (hereafter abbreviated as MS) and a homogenization analysis (hereafter abbreviated as HA).
BACKGROUND OF THE INVENTION
When analyzing the behavior of an engineering structure made of complex mixed materials, a three phase concept of the material profile can be introduced. The first phase of the profile is at a macro-continuum level of the structure. That is, the structure is built as an assembly of macro-continuum elements. The second phase is at a micro-continuum level of the structure. It should be noted that most macro-continuum materials, which seem to be homogeneous in the macro-continuum level, in fact comprise several constituent components when considered from the microscopic point of view. This kind of material is said to be micro-inhomogeneous. The third phase of the profile is at a molecular level. This phase is founded on the physical fact that each constituent material of the micro-continuum consists of a vast number of atoms and/or molecules.
As an example, a bridge is considered as a target structure that is made of granite. The granite is a macro-continuum material. Granite includes three major component minerals such as quartz, feldspar and mica in the micro-continuum level thereof. That is, the micro-structure of granite comprises three constituent minerals, each in turn including a set of atoms/molecules that constitutes a molecular system. Similarly, a concrete dam structure has the same phases. That is, the concrete is a macro-continuum material; gravel, sand and cement paste are micro-continuum materials; and each of the micro-continuum materials constitutes a molecular system.
When designing an engineering structure with plural member elements, it is known to follow a theory of macro-phenomenological mechanics, and to perform a series of experiments in order to determine the macro-mechanical properties of each macro- continuum element. It should be recalled that the material properties obtained in these experiments are averaged in the specimen. This procedure is applied in the same manner in many fields of mechanics such as mechanical engineering and civil engineering.
In order to overcome the theoretical insufficiency involved in the above described macro-phenomenological theory, it is commonly believed that for the micro-inhomogeneous material the size of experimental specimens must be more than ten times larger than the largest size of the constituent components. However, in fact it is not truly known if this belief is true. Furthermore, it is difficult for the macro-phenomenological theory to recognize what happens in the micro-continuum level, although the local phenomena are directly related to the global behavior of the structure.
It can thus be said that the conventional macro-phenomenological procedure is not appropriate for analyzing the behavior of the micro-inhomogeneous material, especially where such a material is used under very extreme engineering condition such as high pressure, high temperature and/or long elapsed time.
In analyzing the behavior of micro-inhomogeneous material two essential problems must be solved. Firstly, it is necessary to determine characteristics of constituent components of the micro-continuum which are directly affected by their molecular movement. Secondly, there must be developed an approach to relating the microscale characteristics to the macroscale behavior of the structure and the macro-continuum elements thereof.
The prior art has not yet succeeded in developing such a fully unified procedure to analyze the molecular movements of the constituent components of the micro-continuum with respect to the macroscale behavior of the structure, much less to design the micro-structure optimally based on such considerations.
MS is a known type of a computer simulation technique. In an MS computation, one gives a material system which consists of particles, atoms and/or molecules, and provides two physical laws, that is, the interatomic interaction potential and the equation of motion or equilibrium. Positions, velocities and/or accelerations of all particles are then calculated under the foregoing physical laws. A statistical thermodynamics procedure is applied to the simulated results, and one can estimate bulk-based physicochemical properties of the material (hereinafter called the bulk properties of the material) such as structure factor of the solid crystal. It is noted that the bulk properties represent thermodynamical averages of the ensemble of particles.
Three classes of MS methods are known:
1) the Monte Carlo method (hereafter abbreviated as MC),
2) the Molecular Mechanics method (hereafter abbreviated as MM), and
3) the Molecular Dynamics method (hereafter abbreviated as MD).
MC, developed by Metropolis et al, estimates the statistical equilibrium state of particles by generating their displacements randomly. Various thermomechanical properties are then calculated by averaging over the states in the Markov chain.
MM is applied for a molecular system which consists of a finite number of atoms, and determines the equilibrium state by optimizing the structure and potential energy. The bulk properties are calculated by using statistical thermodynamics procedure. MM is mainly used in the field of organic chemistry.
On the other hand, MD solves the equation of motion for a system of particles under a given interatomic interaction potential by using a time-discrete finite difference scheme, and the whole time trajectories of particles are specified. The bulk properties are calculated by using statistical thermodynamics procedure for the results.
As MC and MM provide no knowledge of chronological trajectories of particles, these techniques are incapable of considering quantities that are defined in terms of particle motion, such as diffusion. In this sense, except for computational efficiency, MD is more useful so the MD procedure is shown herein as a typical example of MS.
In the MD calculation, the law of conservation of linear momentum is applied for every particle to get the following equation of motion for the i-th component:
m
i
ⅆ
v
i
ⅆ
t
=
F
i
(
1
)
where m
i
is the mass of the i-th particle, v
i
=dr
i
/dt is its velocity at the position r
i
, and the force F
i
is calculated from the potential function U
ij
between two particles by
F
i
=
∑
j
(
i
≠
j
)
F
ij
;
F
ij
=
-
∇
U
ij
.
(
2
)
The MD system usually contains many particles of atoms and/or molecules in a basic cell, and (for simplicity of calculation) the method uses a three dimensional periodic lattice, which is repeated in each direction. Under these conditions one solves a time discrete form of the equation of motion, and the instantaneous position, velocity and acceleration of each particle are specified. Then, using these results and statistical thermodynamic theory, one computes for the material the bulk properties and their change with time, such as density, diffusivity of atoms, molecular vibrations, temperature- and/or pressure-dependent nonlinear elastic moduli, viscosity, heat capacity and heat conductivity.
The interatomic potential function for all atom-atom pairs plays an essential role for the MD calculation. Equation (3) presents a potential function developed by Kawamura so as to reproduce structural and physical properties of several oxide crystals such as quartz, corundum and feldspars properly.
2-body term:
U
ij
(
r
ij
)
=
z
i
z
j
ⅇ
2
4
πϵ
0
r
ij
+
f
0
(
b
i
+
b
j
)
exp
[
a
i
+
a
j
-
r
ij
b
i
+
b
j
]
-
c
i
c
i
r
ij
6
+
D
ij
[
exp
{
-
2
β
ij
(
r
ij
-
r
ij
*
)
}
-
2
exp
{
-
β
ij
(
r
ij
-
r
ij
*
)
}
Ichikawa Yasuaki
Kawamura Katsuyuki
Nakano Masashi
E.R.C. Co., Ltd.
Jones Hugh
Lowe Hauptman & Gilman & Berner LLP
Teska Kevin J.
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