CN111723463B - Numerical prediction method and system for magnetic behavior of monocrystal Ni-Mn-Ga alloy - Google Patents
Numerical prediction method and system for magnetic behavior of monocrystal Ni-Mn-Ga alloy Download PDFInfo
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Abstract
The invention discloses a numerical prediction method and a numerical prediction system for the magnetomotive behavior of a monocrystal Ni-Mn-Ga alloy, wherein the method comprises the following steps: establishing a control equation set describing the magnetomechanical behavior of the single-crystal Ni-Mn-Ga sample, wherein the control equation set comprises a magnetic field equation, a stress balance equation, an internal variable development equation and a twin crystal interface evolution criterion; designing a coupled finite element-boundary element numerical iteration method to solve the control equation set; and predicting the numerical value of the magnetic behavior of the single crystal Ni-Mn-Ga alloy based on the solved control equation set. The invention adopts a coupled finite element-boundary element numerical method, simulates nonlinear response of a sample area by using a finite element method, simulates distribution of magnetic flux on the surface of the sample by using a boundary element method, can obviously reduce the total number of units, improves the calculation efficiency, and can be widely applied to a calculation simulation technology of multi-field coupling mechanical behaviors of novel intelligent metal materials.
Description
Technical Field
The invention relates to a calculation simulation technology of novel intelligent metal material multi-field coupling mechanical behavior, in particular to a numerical prediction method and a numerical prediction system of single crystal Ni-Mn-Ga alloy magnetic behavior.
Background
Single crystal Ni-Mn-Ga alloys are a typical magnetic shape memory alloy. Under the action of external magnetic field or external force load, the strain of the monocrystal Ni-Mn-Ga alloy can reach 6-10%. Under appropriate loading conditions, the strain occurring in the single crystal Ni-Mn-Ga alloy is also reversible, and the response frequency of the material may exceed 1kHz. Because of the unique material characteristics, the monocrystal Ni-Mn-Ga alloy is widely regarded as an ideal material for manufacturing new-generation drivers and sensors, and has wide application prospect in the fields of machinery, medicine, aviation and the like.
The unique material properties of single crystal Ni-Mn-Ga alloys have attracted extensive research interest in the fields of materials, mechanics, automatic control, etc., over the last two decades. On a microscopic scale, intensive studies have been made on the lattice structure and the domain area distribution of the single crystal ni—mn—ga alloy. Research shows that the intrinsic mechanism of deformation of single crystal Ni-Mn-Ga alloy under the action of magnetic field or external force is the martensite variant reorientation inside the material, and the variant reorientation is realized through the nucleation and migration of twin crystal interface inside the sample. In order to deeply understand the magneto-mechanical properties of the monocrystal Ni-Mn-Ga alloy, a foundation is laid for practical application of the monocrystal Ni-Mn-Ga alloy in the industrial field, a great deal of experimental research is carried out by researchers, and different types of theoretical models are built. However, there are many disadvantages in the research in this aspect. Particularly in the aspect of theoretical modeling of the magnetic behavior of a monocrystal Ni-Mn-Ga sample, the existing theoretical model is mostly built based on a localized viewpoint, and the influence of important factors such as configuration, size and boundary constraint of the sample on the magnetic response of the sample cannot be considered. In fact, in order to describe the overall magnetomechanical behavior of the single crystal Ni-Mn-Ga sample, the magnetic field equation, the stress balance equation, the development equation of the internal variables, and the evolution criteria of the twinning interface that the sample satisfies are considered, so as to construct a complete set of theoretical model control equations. In general, the control equation set describing the overall magnetomechanical behavior of a single crystal Ni-Mn-Ga sample is a complex nonlinear multi-field coupling equation set, and contains variables of different nature and equations (or inequalities) of different forms. Even with numerical methods, solving this system of equations is very difficult. At present, the existing research results in the literature are less common in the aspect of numerical simulation of the overall magnetomechanical behavior of a single crystal Ni-Mn-Ga sample.
The most widely used method in the calculation of numerical simulations is the finite element method. However, for single crystal Ni-Mn-Ga samples, the simulation of the magnetomechanical behavior by using a finite element method faces a difficult problem. This is because the single crystal Ni-Mn-Ga sample will be magnetized under the influence of an external magnetic field, thereby exciting the distribution of demagnetizing fields in the interior and surrounding space of the sample. In order to determine the effect of the demagnetizing field on the response of the sample, both the sample area and the surrounding space need to be considered. To ensure accuracy of the numerical calculations, the size of the surrounding space should typically be 5-10 times the size of the sample. This results in a large number of cells in the surrounding space after discretization of the finite element. Significant storage and computational overhead is required to operate these units. Furthermore, with deformation of the sample (variant reorientation), adjustments to the grid of the surrounding space are also required, making the finite element simulation more cumbersome and complex.
Noun interpretation:
magnetomechanical behaviour: namely, the mechanical or physical response of deformation, magnetization, martensite variant reorientation and the like of the single crystal Ni-Mn-Ga alloy under the action of an external magnetic field or an external force load (the magnetic field and the external force can act simultaneously).
Weak form: i.e. a weighted integral version of the differential equation, where the differential is transferred from the dependent variable to the weight function, so that all natural boundary conditions of the problem are also included in the integral version. The weak form discretization can be used for subsequent finite element calculation.
Disclosure of Invention
In order to solve one of the technical problems, the invention aims to provide a numerical prediction method and a numerical prediction system for the magnetic behavior of a monocrystal Ni-Mn-Ga alloy.
The technical scheme adopted by the invention is as follows:
a numerical prediction method for the magnetic behavior of a monocrystal Ni-Mn-Ga alloy comprises the following steps:
Establishing a control equation set describing the magnetomechanical behavior of the single-crystal Ni-Mn-Ga sample, wherein the control equation set comprises a magnetic field equation, a stress balance equation, an internal variable development equation and a twin crystal interface evolution criterion;
designing a coupled finite element-boundary element numerical iteration method to solve the control equation set;
And predicting the numerical value of the magnetic behavior of the single crystal Ni-Mn-Ga alloy based on the solved control equation set.
Further, the magnetic field equation is:
Wherein, B l is the magnetic induction in Lagrangian form related to demagnetizing field H d,/>And ρ=ρ r/J is the density of the material in the configuration.
Further, the stress balance equation includes:
Wherein, For variant region/>And the interface between the surrounding region Ω r′, μ 0 being the permeability in vacuum; hollow square brackets/>Representing the difference between the given physical quantity and the two sides of a curved surface, at/>Upper partAt/>Upper/>Is the nominal stress tensor.
Further, the internal variable development equation is:
Wherein the method comprises the steps of And
Wherein K θ is the magnetic anisotropy constant; f α(αi) is the energy density caused by magnetic domain interactions.
Further, the twin crystal interface evolution criterion is:
Wherein, Is the configurational force applied to the twin crystal interface in the single crystal Ni-Mn-Ga sample; v k is/>Is a speed of configuration of (2); n is/>Direction/>Normal vector of (2); /(I)Is a constant representing the energy dissipation density; /(I)Acting as a retardation during the reorientation of the variants in the forward direction (variant one→variant two) and in the reverse direction (variant two→variant one).
Further, the design coupling finite element-boundary element numerical iteration method solves the control equation set, including:
Designing a numerical iteration scheme for solving the control equation set, and solving the control equation set step by adopting an iteration method according to the numerical iteration scheme;
Acquiring a first weak form of the magnetic field equation and a second weak form of the stress balance equation, wherein a finite element method is used inside the single crystal Ni-Mn-Ga sample region, and a boundary element method is used at the boundary of the single crystal Ni-Mn-Ga sample region;
discretizing the single crystal Ni-Mn-Ga sample according to the first weak form and the second weak form to obtain a unit stiffness matrix.
Further, the design solves a numerical iteration scheme of the control equation set, and according to the numerical iteration scheme, adopts an iteration method to solve the control equation set step by step, including:
Determining an external magnetic field and a mechanical load, and determining initial variant state distribution in a monocrystal Ni-Mn-Ga sample;
solving a variable value of the control equation set according to the external magnetic field, the mechanical load and the initial variant state distribution by adopting a double-loop iteration scheme;
calculating to obtain a configuration force on a twin crystal interface according to the variable value, substituting the configuration force into the twin crystal interface evolution criterion, and detecting the stability of the twin crystal interface;
If the twin crystal interface is stable, ending the solving step; otherwise, the position of the twin crystal interface is adjusted, and the initial variant state distribution is determined and the variable value is calculated.
Further, the second weak form is:
wherein δx is a variation function,
After discretizing the second weak form, the form is:
The invention adopts another technical scheme that:
A numerical prediction system for the magnetomotive behavior of a single crystal Ni-Mn-Ga alloy comprising:
the system comprises an equation establishment module, a control equation set, a parameter analysis module and a parameter analysis module, wherein the equation establishment module is used for establishing a control equation set for describing the magnetomechanical behavior of a single crystal Ni-Mn-Ga sample, and the control equation set comprises a magnetic field equation, a stress balance equation, an internal variable development equation and a twin crystal interface evolution criterion;
the equation calculation module is used for designing a coupled finite element-boundary element numerical value iteration method to solve the control equation set;
and the numerical value prediction module is used for predicting the numerical value of the magnetic behavior of the single crystal Ni-Mn-Ga alloy based on the solved control equation set.
The invention adopts another technical scheme that:
A numerical prediction system for the magnetomotive behavior of a single crystal Ni-Mn-Ga alloy comprising:
At least one processor;
at least one memory for storing at least one program;
the at least one program, when executed by the at least one processor, causes the at least one processor to implement the method described above.
The invention adopts another technical scheme that:
A storage medium having stored therein processor executable instructions which when executed by a processor are for performing the method as described above.
The beneficial effects of the invention are as follows: the invention adopts a coupled finite element-boundary element numerical method, uses the finite element method to simulate the nonlinear response of the sample area, uses the boundary element method to simulate the distribution of magnetic flux on the surface of the sample, can obviously reduce the total number of units and improves the calculation efficiency.
Drawings
FIG. 1 is a schematic diagram showing the steps of a method for predicting the numerical value of the magnetomotive behavior of a single crystal Ni-Mn-Ga alloy according to an embodiment;
FIG. 2 is a flow chart of a numerical iterative algorithm for solving a system of control equations for a single crystal Ni-Mn-Ga sample in an embodiment;
FIG. 3 is a schematic dimensional diagram of a rectangular hexahedral single crystal Ni-Mn-Ga sample in the example;
FIG. 4 is a schematic diagram of the distribution of the modified regions of a rectangular hexahedral single crystal Ni-Mn-Ga sample in the example;
FIG. 5 is a discretized schematic illustration of a single crystal Ni-Mn-Ga sample region in an example;
FIG. 6 is a discretized schematic of the space around a single crystal Ni-Mn-Ga sample in the example;
FIG. 7 is a schematic diagram of evolution curves of the configurational forces at different twin interfaces in the examples;
FIG. 8 is a graphical representation of predicted response curves of single crystal Ni-Mn-Ga samples during cyclic loading in an example;
FIG. 9 is a schematic diagram of effective magnetization distribution of a single crystal Ni-Mn-Ga sample in the example;
FIG. 10 is a graph showing demagnetizing field distribution of a single crystal Ni-Mn-Ga sample in the example;
FIG. 11 is a block diagram of a numerical prediction system of the magnetomotive behavior of a single crystal Ni-Mn-Ga alloy in an embodiment.
Detailed Description
Embodiments of the present invention are described in detail below, examples of which are illustrated in the accompanying drawings, wherein like or similar reference numerals refer to like or similar elements or elements having like or similar functions throughout. The embodiments described below by referring to the drawings are illustrative only and are not to be construed as limiting the invention.
In the description of the present invention, it should be understood that references to orientation descriptions such as upper, lower, front, rear, left, right, etc. are based on the orientation or positional relationship shown in the drawings, are merely for convenience of description of the present invention and to simplify the description, and do not indicate or imply that the apparatus or elements referred to must have a particular orientation, be constructed and operated in a particular orientation, and thus should not be construed as limiting the present invention.
In the description of the present invention, a number means one or more, a number means two or more, and greater than, less than, exceeding, etc. are understood to not include the present number, and above, below, within, etc. are understood to include the present number. The description of the first and second is for the purpose of distinguishing between technical features only and should not be construed as indicating or implying relative importance or implicitly indicating the number of technical features indicated or implicitly indicating the precedence of the technical features indicated.
In the description of the present invention, unless explicitly defined otherwise, terms such as arrangement, installation, connection, etc. should be construed broadly and the specific meaning of the terms in the present invention can be reasonably determined by a person skilled in the art in combination with the specific contents of the technical scheme.
As shown in fig. 1, the present embodiment provides a numerical prediction method for the magnetomotive behavior of a single crystal Ni-Mn-Ga alloy, which includes the following steps:
S1, establishing a control equation set describing the magnetomechanical behavior of a monocrystal Ni-Mn-Ga sample, wherein the control equation set comprises a magnetic field equation, a stress balance equation, an internal variable development equation and a twin crystal interface evolution criterion;
S2, designing a coupled finite element-boundary element numerical value iteration method to solve a control equation set;
s3, predicting the numerical value of the magnetic behavior of the single crystal Ni-Mn-Ga alloy based on the solved control equation set.
Wherein step S2 includes steps S21-S23:
s21, designing a numerical iteration scheme for solving a control equation set;
s22, deducing a weak form of a magnetic field equation and a stress balance equation;
s23, discretizing the monocrystal Ni-Mn-Ga sample, and deducing a unit stiffness matrix.
In the derivation in step S22, the calculation is performed using the finite element method in the sample region, and the calculation is performed using the boundary element method in the space around the sample. The definition of coupled finite element-boundary element is: namely, a finite element method is used in the simulation of the internal physical field of the single crystal Ni-Mn-Ga sample, and a boundary element method is used in the simulation of the demagnetizing field of the space around the single crystal Ni-Mn-Ga sample, so that the two methods are combined to achieve higher calculation accuracy and better calculation efficiency.
The method of the above embodiment is explained in detail below with reference to fig. 2 to 10.
The first step: and (3) establishing a control equation set for describing the whole magnetomechanical behavior of the single crystal Ni-Mn-Ga sample, wherein the control equation set comprises a magnetic field equation, a stress balance equation, an internal variable development equation and a twin crystal interface evolution criterion.
Magnetic field equation: Wherein/> Stress balance equation: /(I)Internal variables α i and/>The development equation:
Wherein, And
Evolution criteria of twinning interface:
Wherein the method comprises the steps of Is the driving force for the modification reorientation, which is the configurational force exerted on the twin interfaces in the sample;
In the above equation model, B l is the magnetic induction in the lagrangian form with respect to the demagnetizing field H d, And ρ=ρ r/J is the density of the material in the current configuration,/>For variant region/>And the interface between the surrounding region Ω r′, μ 0 being the permeability in vacuum; hollow square brackets/>Representing the difference between the two sides of a curved surface of a given physical quantity, whereUpper/>At/>Upper/>Is the nominal stress tensor; k θ is the magnetic anisotropy constant; f α(αi) is the energy density caused by magnetic domain interactions; v k is/>Is a speed of configuration of (2); n isDirection/>Normal vector of (2); /(I)Is a constant representing the energy dissipation density; /(I)Acting as a retardation during the reorientation of the variants in the forward direction (variant one→variant two) and in the reverse direction (variant two→variant one).
And a second step of: and designing a control equation set of a numerical iteration solution model.
The basic idea of the step is to solve the different types of equations (or inequality) in the control equation set step by adopting an iteration method, so that the complexity of numerical calculation of each iteration step is reduced. A flow chart of the numerical iterative algorithm is shown in fig. 2.
Step 1: the evolution of the externally applied magnetic field H a and the mechanical load t a is determined, and the whole loading process is dispersed into a plurality of loading steps. In each step, H a and t a are input parameters;
Step 2: initial variant status distribution in a given sample Or setting the position of a twin crystal interface in the sample; /(I)In the loading process, the initial variant state of the current loading step can be inherited from the last loading step;
Step 3: solving the equation to determine the value of the independent variable in the control system given the external load H a、ta and the fixed variant state distribution n in the sample; adopting a double-loop iteration scheme; solving the Ni-Mn-Ga magnetic force control system in the inner ring, iteratively updating variables psi, q related to demagnetizing field and magnetization vector, Alpha i, the initial value of the position vector x (0) in the inner ring remains unchanged; in the outer ring, solving the force field equation in the step 1 to obtain an updated position vector x (1);
Step 4: the values x, ψ, Alpha i (i, j=1, 2), from/>The expression of (2) is calculated to obtain the configurational force/>, on the twin crystal interface
Step 5: will configure the forceSubstituting the motion rule of the twin crystal interface to inspect the twin crystal interface/>If all twin interfaces in the sample are stable, the algorithm ends; otherwise, the position of the twin interface should be adjusted and the calculation returns to step 2 of the algorithm.
And a third step of: derivation of the weak form of the set of control equations.
Inside the sample region, the weak form of the demagnetizing field equation is:
wherein δψ is an arbitrary variational function;
After discretization, the form is:
At the sample boundary The boundary integral equation of the demagnetizing field equation is:
Wherein, Is the outer region boundary;
Each boundary element centered on a source point ζ The discretized boundary integral equation is satisfied, and the equation is:
the stress balance equation exists only in the sample region, and its weak form is:
Where δx is the variation function, After discretization, the form is:
the cell matrix and cell vector derived from the above are:
Wherein, Is a unit stiffness matrix,/>And/>Region/>, respectivelyCell vectors of the inner and border.
Fourth step: and predicting the magnetomechanical behavior of the single crystal Ni-Mn-Ga sample.
In this example, a rectangular hexahedral single crystal Ni-Mn-Ga sample having a size of 16.0X14.25X14.25 mm 3 is considered as shown in FIG. 3. Edge of sampleIs parallel to the orthogonal basis { e 1,e2,e3 }. During the course of variant reorientation, two variants coexist in the sample, occupying the region/>, respectivelyAnd/>The different mutation regions are defined by the twin interface/>And (5) separating.
Assuming that the applied magnetic field H a is applied along the e 2 axis, an axial compressive load t a is applied to the left and right surfaces of the sample, as shown in FIG. 4.
The effective magnetization vector can be reduced to
Wherein θ isRotation angle of the middle local magnetization vector, α is/>Volume fraction of the medium domains. The expression of θ and α derived by the variational method is:
the expression of the configuration force can be simplified as:
discrete sample areas using tetrahedral units, comprising 1593 nodes and 6144 volume units; inheritance from a volumetric mesh, sample surface Discrete into 1152 triangular units, the sample area is divided into beveled strip-shaped sub-areas as shown in fig. 5 and 6. It is assumed that the redirection of the changes only occurs between regions I -6-I6. In order to develop specific numerical calculations, the material parameters used were:
s＝0.0625,ρMs＝5.64×105A/m,
κ2＝1.50×1011N/m2,κ3＝κ4＝1.52×1011N/m2
κ5＝κ6＝0.43×1011N/m2
FIGS. 7 (a) -7 (c) are plots of evolution of the forces of the configuration on the pair { I -1,I1},{I-3,I3},{I-5,I5 } respectively, showing the results of a comparison of the conventional finite element method and the coupled finite element-boundary element numerical method, obtained by the two numerical algorithms Very close together. Fig. 8 (a) shows the response curve of the magnetic field-magnetization (magnetization component M 2＝Mi·e2) of the Ni-Mn-Ga sample during cyclic loading, and fig. 8 (b) shows the response curve of the magnetic field-strain (axial strain e 11) of the Ni-Mn-Ga sample during cyclic loading, and the coupled finite element-boundary element numerical method can be well simulated. On the graph shown in fig. 8, 8 points a-H are selected. The distribution of effective magnetization and the distribution of demagnetizing field in the Ni-Mn-Ga sample calculated by the coupled finite element-boundary element numerical method are shown in fig. 9 and 10, respectively, corresponding to these points; wherein, fig. 9 (a) -9 (H) are respectively corresponding to the distribution diagrams of the effective magnetization intensity of 8 points a-H, and fig. 9 (a) -9 (H) are respectively corresponding to the distribution diagrams of the demagnetizing field of 8 points a-H.
As shown in fig. 11, the present embodiment further provides a numerical prediction system for the magnetomotive behavior of a single crystal Ni-Mn-Ga alloy, including:
the equation building module is used for building a control equation set for describing the magnetomechanical behavior of the monocrystal Ni-Mn-Ga sample, wherein the control equation set comprises a magnetic field equation, a stress balance equation, an internal variable development equation and a twin crystal interface evolution criterion;
the equation calculation module is used for designing a coupled finite element-boundary element numerical value iteration method to solve a control equation set;
And the numerical value prediction module is used for predicting the numerical value of the magnetic behavior of the single crystal Ni-Mn-Ga alloy based on the solved control equation set.
The numerical prediction system of the magnetic behavior of the single crystal Ni-Mn-Ga alloy can execute the numerical prediction method of the magnetic behavior of the single crystal Ni-Mn-Ga alloy provided by the embodiment of the method, can execute any combination implementation steps of the embodiment of the method, and has corresponding functions and beneficial effects.
The embodiment also provides a numerical prediction system for the magnetomotive behavior of the single crystal Ni-Mn-Ga alloy, which comprises:
At least one processor;
at least one memory for storing at least one program;
the at least one program, when executed by the at least one processor, causes the at least one processor to implement the method described above.
The numerical prediction system of the magnetic behavior of the single crystal Ni-Mn-Ga alloy can execute the numerical prediction method of the magnetic behavior of the single crystal Ni-Mn-Ga alloy provided by the embodiment of the method, can execute any combination implementation steps of the embodiment of the method, and has corresponding functions and beneficial effects.
It is to be understood that all or some of the steps, systems, and methods disclosed above may be implemented in software, firmware, hardware, and suitable combinations thereof. Some or all of the physical components may be implemented as software executed by a processor, such as a central processing unit, digital signal processor, or microprocessor, or as hardware, or as an integrated circuit, such as an application specific integrated circuit. Such software may be distributed on computer readable media, which may include computer storage media (or non-transitory media) and communication media (or transitory media). The term computer storage media includes both volatile and nonvolatile, removable and non-removable media implemented in any method or technology for storage of information such as computer readable instructions, data structures, program modules or other data, as known to those skilled in the art. Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital Versatile Disks (DVD) or other optical disk storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to store the desired information and which can be accessed by a computer. Furthermore, as is well known to those of ordinary skill in the art, communication media typically embodies computer readable instructions, data structures, program modules or other data in a modulated data signal such as a carrier wave or other transport mechanism and includes any information delivery media.
The embodiments of the present invention have been described in detail with reference to the accompanying drawings, but the present invention is not limited to the above embodiments, and various changes can be made within the knowledge of one of ordinary skill in the art without departing from the spirit of the present invention.
Claims (3)
1. A numerical prediction method for the magnetic behavior of a monocrystal Ni-Mn-Ga alloy is characterized by comprising the following steps:
Establishing a control equation set describing the magnetomechanical behavior of the single-crystal Ni-Mn-Ga sample, wherein the control equation set comprises a magnetic field equation, a stress balance equation, an internal variable development equation and a twin crystal interface evolution criterion;
designing a coupled finite element-boundary element numerical iteration method to solve the control equation set;
Predicting the numerical value of the magnetic behavior of the single crystal Ni-Mn-Ga alloy based on the solved control equation set;
The design coupling finite element-boundary element numerical iteration method solves the control equation set, and comprises the following steps:
Designing a numerical iteration scheme for solving the control equation set, and solving the control equation set step by adopting an iteration method according to the numerical iteration scheme;
Acquiring a first weak form of the magnetic field equation and a second weak form of the stress balance equation, wherein a finite element method is used inside the single crystal Ni-Mn-Ga sample region, and a boundary element method is used at the boundary of the single crystal Ni-Mn-Ga sample region;
discretizing a monocrystal Ni-Mn-Ga sample according to the first weak form and the second weak form to obtain a unit stiffness matrix;
the design solves the numerical iteration scheme of the control equation set, adopts an iteration method to solve the control equation set step by step according to the numerical iteration scheme, and comprises the following steps:
determining an applied magnetic field and a mechanical load, and determining initial variant state distribution in the single crystal Ni-Mn-Ga sample;
solving a variable value of the control equation set according to the external magnetic field, the mechanical load and the initial variant state distribution by adopting a double-loop iteration scheme;
calculating to obtain a configuration force on a twin crystal interface according to the variable value, substituting the configuration force into the twin crystal interface evolution criterion, and detecting the stability of the twin crystal interface;
If the twin crystal interface is stable, ending the solving step; otherwise, adjusting the position of the twin crystal interface, and returning to determining the initial variant state distribution and calculating the variable value;
the second weak form is:
wherein δx is a variation function, For variant region,/>For variant region/>And an interface between the surrounding area;
After discretizing the second weak form, the form is:
the cell matrix and cell vector derived from the above are:
Wherein, Is a unit stiffness matrix,/>For volume cell region/>Internal element vector,/>Is the boundary surface unit areaUpper cell vector.
2. A numerical prediction system for the magnetomotive behavior of a single crystal Ni-Mn-Ga alloy, comprising:
the system comprises an equation establishment module, a control equation set, a parameter analysis module and a parameter analysis module, wherein the equation establishment module is used for establishing a control equation set for describing the magnetomechanical behavior of a single crystal Ni-Mn-Ga sample, and the control equation set comprises a magnetic field equation, a stress balance equation, an internal variable development equation and a twin crystal interface evolution criterion;
the equation calculation module is used for designing a coupled finite element-boundary element numerical value iteration method to solve the control equation set;
the numerical value prediction module is used for predicting the numerical value of the magnetic behavior of the single crystal Ni-Mn-Ga alloy based on the solved control equation set;
The design coupling finite element-boundary element numerical iteration method solves the control equation set, and comprises the following steps:
Designing a numerical iteration scheme for solving the control equation set, and solving the control equation set step by adopting an iteration method according to the numerical iteration scheme;
Acquiring a first weak form of the magnetic field equation and a second weak form of the stress balance equation, wherein a finite element method is used inside the single crystal Ni-Mn-Ga sample region, and a boundary element method is used at the boundary of the single crystal Ni-Mn-Ga sample region;
discretizing a monocrystal Ni-Mn-Ga sample according to the first weak form and the second weak form to obtain a unit stiffness matrix;
the design solves the numerical iteration scheme of the control equation set, adopts an iteration method to solve the control equation set step by step according to the numerical iteration scheme, and comprises the following steps:
determining an applied magnetic field and a mechanical load, and determining initial variant state distribution in the single crystal Ni-Mn-Ga sample;
solving a variable value of the control equation set according to the external magnetic field, the mechanical load and the initial variant state distribution by adopting a double-loop iteration scheme;
calculating to obtain a configuration force on a twin crystal interface according to the variable value, substituting the configuration force into the twin crystal interface evolution criterion, and detecting the stability of the twin crystal interface;
If the twin crystal interface is stable, ending the solving step; otherwise, adjusting the position of the twin crystal interface, and returning to determining the initial variant state distribution and calculating the variable value;
the second weak form is:
wherein δx is a variation function,
After discretizing the second weak form, the form is:
the cell matrix and cell vector derived from the above are:
Wherein, Is a unit stiffness matrix,/>For volume cell region/>Internal element vector,/>Is the boundary surface unit areaUpper cell vector.
3. A numerical prediction system for the magnetomotive behavior of a single crystal Ni-Mn-Ga alloy, comprising:
At least one processor;
at least one memory for storing at least one program;
When the at least one program is executed by the at least one processor, the at least one processor implements a numerical prediction method of the magnetomechanical behavior of a single crystal Ni-Mn-Ga alloy according to claim 1.
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