CN102819647A - Finite element modeling method of random microstructure of heterogeneous material - Google Patents
Finite element modeling method of random microstructure of heterogeneous material Download PDFInfo
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Abstract
The invention provides a finite element modeling method of a random microstructure of a heterogeneous material, and particularly relates to a method of establishing a finite element network model of the random microstructure of the heterogeneous material based on microstructure probability distribution information and a random algorithm. The method comprises the following steps of: determining a microstructure probability distribution function by using a real or dummy heterogeneous material component phase physical distribution form; converting a probability distribution function into a discrete space of the model on the basis of establishing the finite element network topological model of the material; and determining material attributes of each unit in the definite element model by using random real numbers generated by a pseudo random number generator and consistently distributed in an interval [0, 1], thereby establishing the random microstructure model of the heterogeneous material. The method is suitable for the heterogeneous materials in different modes; the established definite network model can be directly used for analyzing relation among microscopic property, microscopic structure, macroscopic performance of the heterogeneous material; and basis is provided for development and preparation of a new material.
Description
Technical field
The invention belongs to the finite element modeling technical field, relate to a kind of finite element modeling method, especially a kind of heterogeneous material random microscopic structure finite element modeling method.
Background technology
The macro property (like rigidity, intensity and toughness etc.) of the heterogeneous material (like compound substance, porosint etc.) that is made up of various ingredients is mainly by its micromechanism decision; Therefore, the micromechanism of thoroughly studying heterogeneous material to the influence of its macro property for design with to develop new high-performance heterogeneous material significant.Finite Element Method is to concern one of effective method between research material micromechanism and macro property, and this method need at first be set up the micromechanism finite element model of heterogeneous material.For the material that development and preparation is come out, we can reconstruct the finite element grid model of its true micromechanism of reflection on the basis of gathering the tomography X image, and the macro-mechanical property of this material that is used to calculate to a nicety.Yet; In order to set up the relation between heterogeneous material micromechanism and its macro property; Micromechanism information should appear in the research process as a variable; But receive the restriction of experimental cost and researchist's energy, development and preparation is unpractical with the multiple material of single micromechanism information change.Development can reflect that the finite element modeling method of heterogeneous material micromechanism random variation helps to overcome this defective, and foundation is provided for the design of novel heterogeneous material.
Summary of the invention
The objective of the invention is to overcome the shortcoming of above-mentioned prior art; A kind of heterogeneous material random microscopic structure finite element modeling method is provided; This method is the basis with real or imaginary heterogeneous material component phase physical distribution form; Set up its random microscopic structural model according to the micromechanism probability distribution function of heterogeneous material, its help to set up heterogeneous material micromechanism and macro property to stride yardstick related and explore the optimum micromechanism of macro property.
The objective of the invention is to solve through following technical scheme:
This heterogeneous material random microscopic structure finite element modeling method may further comprise the steps:
1) confirms the probability distribution function of component phase according to the microstructure features of each component phase of heterogeneous material;
2) set up the finite element grid topological model of heterogeneous material;
3) probability distribution function is transformed in the discrete space, and confirms the material properties of each unit in the finite element grid model by random algorithm.
Further, above-mentioned steps 1) specifically carry out according to following method:
Shape and distribution thereof that each component of heterogeneous material is assembled bunch mutually have specific forms, and promptly the probability that in the universe space of material, occurs of each component can be expressed with specific mathematical distribution function; For the equally distributed at random M phase heterogeneous material of each component phase, the volume fraction of each phase material is respectively v
n(n=1,2 ..., M), then its probability distribution function is v
n(X)=v
n(n=1,2 ..., M), corresponding cumulative distribution function does
For two phase heterogeneous materials of micromechanism distribution gradient, the component x that hands down
kThe direction distribution gradient, then its probability distribution function is respectively v
1(X)=2v
1/ (1+exp (g-2gx
k/ X
k)) and v
2(X)=1-v
1(X), g is the gradient index in the formula, and the bigger then variable gradient between the two component phase materials of this value is just big more, X
kFor the material monolithic model along x
kThe overall dimensions of direction.
Further, above-mentioned steps 2) specifically carry out according to following method:
At first confirm number of unit W, H and T and unit size w, h and the t of finite element model along x, y and z direction; Set up the finite element grid topological model that is made up of eight node rectangular parallelepiped unit or four node rectangular elements then, the coordinate of n node is confirmed by following two groups of expression formulas respectively in the three peacekeeping two dimensional models:
In the formula " % " for division of integer get surplus, the maximum integer that
gets the less-than operation object; The node of n unit is respectively in the model:
Further, above-mentioned steps 3) specifically carry out according to following method:
Adopt Mersenne Twister and Mitchell-Moore pseudorandom number generator to generate required random number in the micromechanism modeling, in order to improve initial random degree, the random seed generating algorithm below adopting:
Seed in the formula [n] (n=1,2 ...) be the seed of randomizer, number seeds is by concrete pseudorandom number generator decision, t
SBe the current time of computer system, t
BAnd t
EBe respectively the system time when program began and finished the last period, p is the number of bits factor, by the unit scale of micromechanism modeling confirm and " be respectively step-by-step with and accord with to shift operation;
The continuous probability distribution function of heterogeneous material component phase is transformed in the discrete space of finite element grid model, confirms the material properties of each unit in the finite element model by the consistent R of real number at random that is distributed in the interval [0,1] of pseudorandom number generator generation; For the heterogeneous heterogeneous material of micromechanism stochastic distribution, the material properties of n unit is confirmed by following formula in the finite element model:
S in the formula
nBe probability distribution function v
1(X) the discrete value at n place in the unit afterwards, that is:
I in the formula
kFor model along x
kThe unit total number of direction, i
kFor unit n along x
kThe discrete coordinates of direction is for three peacekeeping two dimensional models, i
m(m=1,2,3) are respectively:
The present invention has following beneficial effect:
The present invention provides a kind of economy effective method for the model of setting up microstructure features random variation such as each component phase distribution form of heterogeneous material, state of aggregation; This method is applicable to multi-form heterogeneous material, comprises heterogeneous material, FGM and porous medium etc.; The finite element grid model that this method is set up can directly be used for analyzing the relation between micromechanism such as form, aggregation characteristic and the distribution of each component phase of heterogeneous material and the macro property; The physical distribution form of each component phase of heterogeneous material can be confirmed that by real material this helps to study the relation of current material micromechanism and macro property, for the improvement of material property provides foundation; The physical distribution form of each component phase also can freely design with fabricating, thereby explores the optimum micromechanism of macro property, for the development and preparation of new material provides reference.
Description of drawings
Fig. 1 process flow diagram of the present invention;
The foundation of Fig. 2 two-dimensional finite unit network topology model;
The foundation of Fig. 3 three-dimensional finite element mesh topological model;
Fig. 4 two phase heterogeneous material micromechanisms are the two-dimensional finite unit grid model of stochastic distribution;
Fig. 5 three-phase heterogeneous material micromechanism is the three-dimensional finite element mesh model of stochastic distribution;
The two-dimensional finite unit grid model of Fig. 6 two phase heterogeneous material micromechanism distribution gradient;
The three-dimensional finite element mesh model of Fig. 7 two phase heterogeneous material micromechanism distribution gradient.
Embodiment
Below in conjunction with accompanying drawing the present invention is done and to describe in further detail:
The present invention confirms on the basis of probability distribution function in the physical distribution form of utilizing each component phase of heterogeneous material; Set up the finite element grid topological model of heterogeneous material; And confirm the material properties of each unit through Mitchell-Moore and Mersenne-Twister random algorithm, set up the random microscopic structural model of heterogeneous material thus.The practical implementation flow process of this method is as shown in Figure 1, describes concrete technical matters in detail according to this flow process below.
1. confirm the probability distribution function of component phase according to the microstructure features of each component phase of heterogeneous material:
Shape and distribution thereof that each component of heterogeneous material is assembled bunch mutually have specific forms, and promptly the probability that in the universe space of material, occurs of each component can use specific mathematical distribution function to express.The material that comes out for development and preparation; Can confirm the probability distribution function of each component phase according to tomographic image or through a series of measures such as metallographic methods; Also can fabricate some micromechanism distribution forms; And construct the probability distribution function of each component phase thus, can make the material macro property reach optimum micromechanism form thereby explore.
The present invention explains component confirming of probability distribution function mutually with the heterogeneous material of micromechanism stochastic distribution and Gradient distribution.Assemble a bunch equally distributed at random heterogeneous material mutually for each component, the volume fraction of establishing its each phase material is respectively v
n(n=1,2 ..., M), wherein M is the component phase number of material, then n component is v at the probability that an X place occurs
n(X)=v
n, promptly the probability distribution function of each component of this material in the universe space is v
n(X)=v
n(n=1,2 ..., M), corresponding cumulative distribution function does
For two phase heterogeneous materials of micromechanism distribution gradient, its component is assembled a certain direction distribution gradient in bunch edge mutually, and still is at random evenly distribution along other directions.If the volume fraction of each phase material is respectively v
n(n=1,2), material is along x
kThe direction distribution gradient, then the probability distribution function of each phase material is respectively v
1(X)=2v
1/ (1+exp (g-2gx
k/ X
k)) and v
2(X)=1-v
1(X), in the formula, g is the gradient index, and the bigger then variable gradient between the two component phase materials of this value is just big more; X
kFor the material monolithic model along x
kThe overall dimensions of direction.
2. set up the finite element grid topological model of heterogeneous material:
(1) two dimensional model
At first confirm number of unit W and H and unit size w and the h of two-dimensional finite element model, set up the finite element grid topological model that constitutes by four node rectangular elements as shown in Figure 2 then along x and y direction according to the target of heterogeneous material random microscopic structural modeling.According to the numbering of node and unit among the figure, the coordinate of n node can be confirmed by following mathematical relation simply in the model
In the formula, x
nAnd y
nBe respectively the coordinate of node n along x and y direction, " % " for division of integer get surplus,
Get the maximum integer of less-than operation object.N unit confirmed by following formula along four nodes counterclockwise arranging in the model
(2) three-dimensional model
Equally obtain number of unit T and the unit size t of three-dimensional finite element model along the z direction according to the modeling target, and set up finite element grid topological model as shown in Figure 3, this model is made up of the rectangular parallelepiped unit of eight nodes.According to the node and the element number mode of model among the figure, the coordinate that can obtain n node in the model does
N unit confirmed by following formula along eight nodes counterclockwise arranging in the model
3. micromechanism modeling
(1) generation of random number
The present invention adopts degree of randomness higher, and Mersenne Twister and Mitchell-Moore pseudorandom number generator that the cycle is long generate required random number in the micromechanism modeling, in order to improve initial random degree, and the random seed generating algorithm below adopting
In the formula, seed [n] (n=1,2 ...) be the seed of randomizer, number seeds is by concrete pseudorandom number generator decision; t
SBe the current time of computer system; t
BAnd t
EBe respectively the system time when program began and finished the last period; P is the number of bits factor, is confirmed by the unit scale of micromechanism modeling; & with " be respectively step-by-step with and accord with to shift operation.
(2) the model unit attribute confirms
The continuous probability distribution function of heterogeneous material component phase is transformed in the discrete space of finite element grid topological model; Just can confirm the material properties of each unit in the finite element model by the consistent real number at random that is distributed in the interval [0,1] that pseudorandom number generator produces.For the heterogeneous heterogeneous material of micromechanism stochastic distribution, the material properties of n unit is confirmed by following formula in the finite element model:
In the formula, the consistent at random real number that be distributed in interval [0,1] of R for producing by pseudorandom number generator.
For two phase heterogeneous materials of micromechanism distribution gradient, the material properties of n unit is confirmed by following formula in its finite element model:
In the formula, s
nBe probability distribution function v
1(X) the discrete value at n place in the unit afterwards, promptly
In the formula, I
kFor finite element model along x
kThe unit total number of direction, i.e. I
1=W, I
2=H, I
3=T; i
kFor unit n in the finite element model along x
kThe discrete coordinates of direction, for two dimensional model, i
m(m=1,2) are respectively:
And for three-dimensional model, i
m(m=1,2,3) are respectively:
4. embodiment
(1) micromechanism stochastic distribution model
Get parameter W=H=50, w=h=0.1, v
1=0.6, v
2=0.4 just can set up the two-dimensional finite unit grid model that two phase heterogeneous material micromechanisms as shown in Figure 4 are stochastic distribution, and in the model, black region is volume fraction v
1=0.6 component phase, white portion are volume fraction v
1=0.4 component phase; Get parameter W=H=50, T=20, w=h=t=0.1, v
1=0.5, v
2=0.3, v
3=0.2 can set up the three-dimensional finite element mesh model that as shown in the figure 5 three-phase heterogeneous material micromechanism is stochastic distribution, among the figure, and the component phase that the region representation volume fraction that gray scale is big more is more little.
(2) micromechanism Gradient distribution model
Get parameter W=H=50, w=h=0.1, v
1=0.6, v
2=0.4, g=5, x
k=x just can set up the two-dimensional finite unit grid model of two phase heterogeneous material micromechanism distribution gradient as shown in Figure 6, and in the model, black region is volume fraction v
1=0.6 component phase, white portion are volume fraction v
1=0.4 component phase, the transition between two phase materials is comparatively level and smooth; Get parameter W=H=50, T=20, w=h=t=0.1, v
1=0.7, v
2=0.3, g=20, x
k=y can set up the three-dimensional finite element mesh model of two phase heterogeneous material micromechanism distribution gradient as shown in Figure 7, and among the figure, black region is volume fraction v
1=0.7 component phase, white portion are volume fraction v
1=0.3 component phase, the gradient between two phase materials sharply changes.
Application of the present invention is not limited in above modeled example; Know-why by preamble is addressed can be known; It is applicable on the one hand sets up the different two and three dimensions finite element grid model of component phase volume fraction to any heterogeneous heterogeneous material, is applicable on the other hand micromechanism is set up component phase volume fraction and the different two and three dimensions finite element grid model of gradient along any two phase heterogeneous materials of a certain direction distribution gradient.
Claims (4)
1. a heterogeneous material random microscopic structure finite element modeling method is characterized in that, may further comprise the steps:
1) confirms the probability distribution function of component phase according to the microstructure features of each component phase of heterogeneous material;
2) set up the finite element grid topological model of heterogeneous material;
3) probability distribution function is transformed in the discrete space, and confirms the material properties of each unit in the finite element grid model by random algorithm.
2. heterogeneous material random microscopic structure finite element modeling method according to claim 1 is characterized in that said step 1) is specifically carried out according to following method:
Shape and distribution thereof that each component of heterogeneous material is assembled bunch mutually have specific forms, and promptly the probability that in the universe space of material, occurs of each component can be expressed with specific mathematical distribution function; For the equally distributed at random M phase heterogeneous material of each component phase, the volume fraction of each phase material is respectively v
n(n=1,2 ..., M), then its probability distribution function is v
n(X)=v
n(n=1,2 ..., M), corresponding cumulative distribution function does
For two phase heterogeneous materials of micromechanism distribution gradient, the component x that hands down
kThe direction distribution gradient, then its probability distribution function is respectively v
1(X)=2v
1/ (1+exp (g-2gx
k/ X
k)) and v
2(X)=1-v
1(X), g is the gradient index in the formula, and the bigger then variable gradient between the two component phase materials of this value is just big more, X
kFor the material monolithic model along x
kThe overall dimensions of direction.
3. heterogeneous material random microscopic structure finite element modeling method according to claim 1 is characterized in that said step 2) specifically carry out according to following method:
At first confirm number of unit W, H and T and unit size w, h and the t of finite element model along x, y and z direction; Set up the finite element grid topological model that is made up of eight node rectangular parallelepiped unit or four node rectangular elements then, the coordinate of n node is confirmed by following two groups of expression formulas respectively in the three peacekeeping two dimensional models:
In the formula " % " for division of integer get surplus, the maximum integer that
gets the less-than operation object; The node of n unit is respectively in the model:
4. heterogeneous material random microscopic structure finite element modeling method according to claim 1 is characterized in that said step 3) is specifically carried out according to following method:
Adopt Mersenne Twister and Mitchell-Moore pseudorandom number generator to generate required random number in the micromechanism modeling, in order to improve initial random degree, the random seed generating algorithm below adopting:
Seed in the formula [n] (n=1,2 ...) be the seed of randomizer, number seeds is by concrete pseudorandom number generator decision, t
SBe the current time of computer system, t
BAnd t
EBe respectively the system time when program began and finished the last period, p is the number of bits factor, by the unit scale of micromechanism modeling confirm and " be respectively step-by-step with and accord with to shift operation;
The continuous probability distribution function of heterogeneous material component phase is transformed in the discrete space of finite element grid model, confirms the material properties of each unit in the finite element model by the consistent R of real number at random that is distributed in the interval [0,1] of pseudorandom number generator generation; For the heterogeneous heterogeneous material of micromechanism stochastic distribution, the material properties of n unit is confirmed by following formula in the finite element model:
S in the formula
nBe probability distribution function v
1(X) the discrete value at n place in the unit afterwards, that is:
I in the formula
kFor model along x
kThe unit total number of direction, i
kFor unit n along x
kThe discrete coordinates of direction is for three peacekeeping two dimensional models, i
m(m=1,2,3) are respectively:
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