CN116595684B - Topological optimization design method for embedded piezoelectric driving compliant mechanism based on size constraint - Google Patents
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Abstract
The invention provides a topological optimization design method of an embedded piezoelectric driving compliant mechanism based on size constraint, which relates to the technical field of compliant mechanisms and comprises the following steps: defining a design domain, initializing design parameters, adopting a point density-level set joint topology description model, utilizing a level set function to represent piezoelectric drivers of any shape, adopting an independent point density interpolation method to obtain density field distribution of a main structure, maximizing output displacement of a compliant mechanism to be an objective function, taking the volume fraction of the main structure of the compliant mechanism, the fact that the piezoelectric drivers do not interfere with the design domain and the minimum size design are taken as constraint conditions, establishing an embedded piezoelectric driving compliant mechanism topology optimization model considering size constraint, and solving a topology optimization problem through a moving progressive line algorithm. According to the invention, the minimum size of the embedded movable piezoelectric driving compliant mechanism is controlled by adding the size constraint, and the finally obtained target configuration can effectively realize the required minimum size, so that the practical process manufacturing is facilitated.
Description
Technical Field
The invention relates to the technical field of optimization design of compliant mechanisms, in particular to a topological optimization design method of an embedded piezoelectric driving compliant mechanism based on size constraint.
Background
A compliant mechanism is a mechanism that effects movement, force transfer, and energy conversion through deformation of its own material. Compared with the traditional rigid mechanism, the flexible mechanism has the advantages of easy processing, no assembly, no need of lubrication, high integration level and the like, so that the flexible mechanism has wide application prospects in the fields of micro-electromechanical systems, precise positioning, micro-nano manufacturing and the like.
In engineering structural design issues, many structural systems often require embedding one or more components of a particular shape into a limited design space to achieve certain functional design requirements. Therefore, it is necessary to not only find the optimal positions and placement directions of the embedded components in the allowed design space throughout the design process, but also design the support structures connecting the embedded components to improve the overall performance of the structure.
The traditional design method is mainly divided into two steps, namely component layout design and structure optimization design. The former is mainly to optimize the position and placement direction of each component in the fixed area, while the latter is to perform the optimization design of the bearing structure according to the target after determining the position and direction of the component, where the component area is usually set as an un-designable area. The step design approach cuts off the internal links between the component layout and the load-bearing structure and fails to reach an optimal overall performance design.
Disclosure of Invention
Aiming at the defects of the prior art, the invention aims to provide a topological optimization design method of an embedded piezoelectric driving compliant mechanism based on size constraint, which aims to solve the technical problem of poor overall performance design in the prior art.
In order to achieve the above object, the present invention is achieved by the following technical scheme: the topological optimization design method of the embedded piezoelectric driving compliant mechanism based on the size constraint is characterized by comprising the following steps of:
step S1, defining a design domain, a fixed boundary condition, a punishment coefficient, a unit filtering radius, a virtual load size, an elastic modulus of two materials of a main body structure and an embedded component, a Poisson ratio and volume constraint of the two materials of the main body structure and the embedded component, applying virtual spring stiffness of an output end, setting the shape, the size and the position of a piezoelectric driver, applying voltage, dispersing the design domain into plane 4 node units, and initializing a unit node density design variable;
step S2, interpolating the unit node density design variables by using a rational interpolation model, calculating to obtain the density values of each point in the design domain by using the rational interpolation model, and representing the topological configuration of the main structure;
step S3, independently describing a movable piezoelectric driver by adopting a level set function, dispersing a design domain of the level set function into grids with the same size as the plane 4 node units, and expressing the position, the direction and the speed of the piezoelectric driver in the design domain by using the level set function;
s4, adopting a filtering threshold topology optimization scheme, utilizing the design domain, the filtering design domain and the physical domain as calculation frames, constructing a structure indication function to respectively represent a solid phase and a space phase of the main structure, and referring to two dimension constraints to respectively realize the minimum dimension control of the solid phase and the space phase;
s5, taking the volume fraction and the size of the materials of the main structure and the fact that the piezoelectric driver does not interfere with a design domain as constraint conditions, taking the maximization of the output displacement of the compliant mechanism as an objective function, and establishing a topological optimization design model of the embedded movable piezoelectric driving compliant mechanism based on size constraint;
s6, solving the output displacement, the non-overlapping control of the piezoelectric driver and the minimum size and volume constraint of the main structure by using the topological optimization design model of the embedded movable piezoelectric driving compliant mechanism based on the size constraint, solving the sensitivity of the output displacement, the non-overlapping control and the volume constraint of the main structure of the objective function mechanism by using a chained derivative method, and solving the sensitivity of the minimum size constraint by using a local gradient derivative method;
s7, updating design variables by using a moving asymptotic algorithm, solving a Hamilton-Jacobi equation by using the obtained virtual speed variables, updating a level set function, and judging whether convergence conditions are met;
if the convergence condition is not satisfied, turning to step S2;
and step S71, if the convergence condition is met, ending the topology optimization process, and obtaining the topology configuration of the embedded movable piezoelectric driving compliant mechanism meeting the minimum size control.
According to an aspect of the foregoing technical solution, the step S2 specifically includes:
step S21, calculating to obtain the density value of each point in the design domain by using the following calculation formula and using a shepherd interpolation model:
in the formula ,for the pseudo-density of the ith cell in the body structure,/or->For shepiad interpolation function, +.>Designing a variable number for the point density for interpolation;
step S22, the Young modulus of each node in the design domain is obtained by adopting the following calculation formula and based on the sheplate interpolation model:
in the formula ,is the Young's modulus of the material of the body structure, < >>Is a penalty coefficient.
According to an aspect of the foregoing technical solution, the step S3 specifically includes:
s31, discretizing a level set design domain into finite element grids with the same size as the density interpolation model;
step S32, describing boundaries of the piezoelectric driver by applying a level set function:
wherein ,representing the inner area of the piezo-electric actuator, < >>Representing the boundaries of the piezoelectric actuator.
According to an aspect of the foregoing technical solution, the step S4 specifically includes:
step S41, defining a design domainFilter design field->And physical Domain->And said design domain, said filtering design domain +.>And the following formula is satisfied between the physical domains:
wherein ,is a unit->Neighborhood set in the filtering domain of (2) ->Is the radius of the linear cap filter, +.>Is a unit->Volume of->Is a unit-> and />Center coordinates of +.> and />A weighting function of the distance between them,/->Control the steepness of the approximate Heaviside function,/->Is a critical value.
According to an aspect of the foregoing solution, after step S41, the method further includes:
step S42, defining inflection regions, namely, an area of 1, an area of 2, and respectively constructing two structural index functions representing a solid phase and a space phase of the main structural material:
wherein, superscript and />Respectively representing a solid phase and an empty phase;
step S43, constructing the following two geometric constraints to achieve minimum size control of the solid and air phases, respectively:
wherein ,for discretizing the collection->The total number of elements in (a) satisfies these two constraints, the value of the filtered field will be greater than the threshold value +_1 at the inflection point +_1>A threshold value +.about.2 at less than +.>。
According to an aspect of the foregoing technical solution, in step S5, the topology optimization design model of the embedded movable piezoelectric driving compliant mechanism based on size constraint is:
wherein ,is a point density design variable for describing the distribution of flexible structural material,/->Is a virtual speed design variable for representing the motion of the piezo actuator,/->Is the +.>The pseudo-density of the individual cells is,describes the translational and rotational angular speeds of the piezo-actuator,/-> and />Representing the structural pseudo-density design variable and the number of piezo-electric actuators, respectively, +.>For output displacement;
representing the volume of material of the body structure; />Volume of material of non-piezoelectric actuator region, +.>Is the volume of the design domain, +.>Is the volume of the piezo-actuator, +.> and />Representing the control of the dimensions of the main structure and the empty phase, respectively.
According to an aspect of the foregoing technical solution, the step S6 specifically includes:
step S61, calculating the sensitivity of the objective function, the volume and the non-overlapping constraint of the compliant mechanism to the design variable by applying an analytical shape sensitivity analysis method;
wherein the finite element form of the objective function is expressed as:the derivative of the objective function to the point density design variable is: />Volume constraint->For the firstThe derivative of the individual point density design variables is: />；
The shape sensitivity of the objective function is:
；
wherein ,the shape derivative representing the virtual velocity perturbation vector of the piezoelectric actuator, the non-overlapping constraint is:
。
according to an aspect of the foregoing solution, after step S61, the method further includes:
step S62, obtaining the derivative of the objective function and the constraint function on the virtual speed design variable through a chain derivation rule:
；
wherein ,is->The partial derivatives of the objective function and the constraint function, respectively.
According to an aspect of the foregoing solution, after step S62, the method further includes:
and step S63, solving the sensitivity of the minimum size constraint by using a local gradient derivative method, filtering the density of the control points by using a density filtering method, and obtaining the density of the filtered control points by using a weighted average method of the densities of adjacent control points.
According to an aspect of the foregoing technical solution, in step S7, the step of solving the Hamilton-Jacobi equation from the obtained virtual speed variable specifically includes:
step S72, updating design variables by adopting a moving progressive line algorithm;
step S73, solving a Hamilton-Jacobi equation based on the virtual speed variable:
in the formula ,for normal speed +.>Representing the velocity of movement of a point on the boundary of the piezoelectric actuator,representing the normal direction vector of the corresponding point.
The invention has the beneficial effects that: the method comprises the steps of adopting a point density-level set joint topology description model, utilizing a level set function to represent piezoelectric drivers of any shape, adopting an independent point density interpolation method to obtain density field distribution of a main structure, maximizing output displacement of a compliant mechanism to be an objective function, taking the volume fraction of the main structure of the compliant mechanism, no interference between the piezoelectric drivers and a design domain and minimum dimension design as constraint conditions, establishing an embedded piezoelectric driving compliant mechanism topology optimization model considering dimension constraint, solving a topology optimization problem through a moving progressive line algorithm, and under the frame of the point density-level set joint topology description, optimizing and placing the piezoelectric drivers each time to influence rigidity of the main structure, wherein the method is different from the traditional method; at the same time, changing the topology of the structure will also affect the next movement of the piezoelectric actuator. The two are mutually influenced and jointly act. The method fundamentally eliminates the defect of step-by-step optimization of the traditional design method, and can simultaneously realize the optimal layout of the piezoelectric driver and the optimal design of the bearing structure. According to the invention, the minimum size of the embedded movable piezoelectric driving compliant mechanism is controlled by adding the size constraint, and the finally obtained target configuration can effectively realize the required minimum size, so that the practical process manufacturing is facilitated.
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The foregoing and/or additional aspects and advantages of the invention will become apparent and may be better understood from the following description of embodiments taken in conjunction with the accompanying drawings in which:
FIG. 1 is a flow chart of a topology optimization design method of an embedded piezoelectric driving compliant mechanism based on size constraint in an embodiment of the invention;
FIG. 2 is a schematic diagram illustrating exemplary displacement inverting amplifier design domains, virtual loads, boundary conditions, and initial positions and sizes of a movable piezoelectric actuator according to an embodiment of the present invention;
FIG. 3 is a schematic diagram of a displacement inverting amplifier topology obtained in an embodiment of the invention;
FIG. 4 is a schematic diagram of a displacement inverting amplifier topology obtained in an embodiment of the invention with minimal dimensional control;
FIG. 5 is an iterative process for displacing the volume fraction and output displacement of an inverting amplifier in accordance with one embodiment of the invention;
FIG. 6 is a basic flow chart of a topology optimization design method for an embedded piezoelectric driven compliant mechanism based on size constraints in an embodiment of the invention.
Detailed Description
In order that the invention may be readily understood, a more complete description of the invention will be rendered by reference to the appended drawings. Various embodiments of the invention are shown in the drawings. This invention may, however, be embodied in many different forms and should not be construed as limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete.
It will be understood that when an element is referred to as being "mounted" on another element, it can be directly on the other element or intervening elements may also be present. When an element is referred to as being "connected" to another element, it can be directly connected to the other element or intervening elements may also be present. The terms "vertical," "horizontal," "left," "right," and the like are used herein for illustrative purposes only.
Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. The terminology used herein in the description of the invention is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. The term "and/or" as used herein includes any and all combinations of one or more of the associated listed items.
Referring to fig. 1, a flow chart of a topology optimization design method of an embedded piezoelectric driving compliant mechanism based on size constraint in a first embodiment of the invention is shown, comprising the following steps:
step S1, defining a design domain, a fixed boundary condition, a punishment coefficient, a unit filtering radius, a virtual load size, an elastic modulus of two materials of a main body structure and an embedded component, a Poisson ratio and volume constraint of the two materials of the main body structure and the embedded component of the compliant mechanism, applying virtual spring stiffness of an output end, setting the shape, the size and the position of a piezoelectric driver, applying voltage, dispersing the design domain into planar 4-node units, and initializing a unit node density design variable.
And S2, interpolating the unit node density design variable by adopting a rational interpolation model, calculating to obtain the density value of each point in the design domain by using the rational interpolation model, and representing the topological configuration of the main structure.
In some embodiments, the step S2 specifically includes:
step S21, calculating to obtain the density value of each point in the design domain by using the following calculation formula and using a shepherd interpolation model:
in the formula ,for the pseudo-density of the ith cell in the body structure,/or->For shepiad interpolation function, +.>Designing a variable number for the point density for interpolation;
step S22, the Young modulus of each node in the design domain is obtained by adopting the following calculation formula and based on the sheplate interpolation model:
in the formula ,is the Young's modulus of the material of the body structure, < >>Is a penalty coefficient.
And S3, independently describing the movable piezoelectric driver by adopting a level set function, dispersing a design domain of the level set function into grids with the same size as the plane 4 node units, and expressing the position, the direction and the speed of the piezoelectric driver moving in the design domain by using the level set function.
In some embodiments, the step S3 specifically includes:
s31, discretizing a level set design domain into finite element grids with the same size as the density interpolation model;
step S32, describing boundaries of the piezoelectric driver by applying a level set function:
wherein ,representing the inner area of the piezo-electric actuator, < >>Representing the boundaries of the piezoelectric actuator.
And S4, adopting a filtering threshold topology optimization scheme, utilizing the design domain, the filtering design domain and the physical domain as a calculation frame, constructing a structure indication function to respectively represent the solid phase and the air phase of the main structure, and referring to two dimension constraints to respectively realize the minimum dimension control of the solid phase and the air phase.
In some embodiments, the step S4 specifically includes:
step S41, defining a design domainFilter design field->And physical Domain->And said design domain, said filtering design domain +.>And the following formula is satisfied between the physical domains:
wherein ,is a unit->Neighborhood set in the filtering domain of (2) ->Is the radius of the linear cap filter, +.>Is a unit->Volume of->Is a unit-> and />Center coordinates of +.> and />A weighting function of the distance between them,/->Control the steepness of the approximate Heaviside function,/->Is a critical value.
Step S42, defining inflection regions, namely, an area of 1, an area of 2, and respectively constructing two structural index functions representing a solid phase and a space phase of the main structural material:
wherein, superscript and />Respectively representing a solid phase and an empty phase;
step S43, constructing the following two geometric constraints to achieve minimum size control of the solid and air phases, respectively:
wherein ,for discretizing the collection->The total number of elements in (a) satisfies these two constraints, the value of the filtered field will be greater than the threshold value +_1 at the inflection point +_1>A threshold value +.about.2 at less than +.>。
And S5, taking the volume fraction and the size of the materials of the main body structure and the fact that the piezoelectric driver does not interfere with a design domain as constraint conditions, taking the output displacement maximization of the compliant mechanism as an objective function, and establishing a topological optimization design model of the embedded movable piezoelectric driving compliant mechanism based on size constraint.
In some embodiments, the topology optimization design model of the embedded movable piezoelectric driving compliant mechanism based on the size constraint in the step S5 is:
wherein ,is a point density design variable for describing the distribution of flexible structural material,/->Is a virtual speed design variable for representing the motion of the piezo actuator,/->Is the +.>The pseudo-density of the individual cells is,describes the translational and rotational angular speeds of the piezo-actuator,/-> and />Respectively representing the pseudo density of the structureDesign variables and number of piezo-electric drives, +.>For output displacement;
representing the volume of material of the body structure; />Volume of material of non-piezoelectric actuator region, +.>Is the volume of the design domain, +.>Is the volume of the piezo-actuator, +.> and />Representing the control of the dimensions of the main structure and the empty phase, respectively.
And S6, solving the output displacement, the non-overlapping control of the piezoelectric driver and the minimum size and volume constraint of the main structure by using the topological optimization design model of the embedded movable piezoelectric driving compliant mechanism based on the size constraint, solving the sensitivity of the output displacement, the non-overlapping control and the volume constraint of the main structure of the objective function mechanism by using a chained derivative method, and solving the sensitivity of the minimum size constraint by using a local gradient derivative method.
In some embodiments, the step S6 specifically includes:
step S61, calculating the sensitivity of the objective function, the volume and the non-overlapping constraint of the compliant mechanism to the design variable by applying an analytical shape sensitivity analysis method;
wherein the finite element form of the objective function is expressed as:the derivative of the objective function to the point density design variable is: />Volume constraint->For->The derivative of the individual point density design variables is: />；
The shape sensitivity of the objective function is:
；
wherein ,the shape derivative representing the virtual velocity perturbation vector of the piezoelectric actuator, the non-overlapping constraint is:
。
step S62, obtaining the derivative of the objective function and the constraint function on the virtual speed design variable through a chain derivation rule:
；
wherein ,is->The partial derivatives of the objective function and the constraint function, respectively.
And step S63, solving the sensitivity of the minimum size constraint by using a local gradient derivative method, filtering the density of the control points by using a density filtering method, and obtaining the density of the filtered control points by using a weighted average method of the densities of adjacent control points.
S7, updating design variables by using a moving asymptotic algorithm, solving a Hamilton-Jacobi equation by using the obtained virtual speed variables, updating a level set function, and judging whether convergence conditions are met;
if the convergence condition is not satisfied, turning to step S2;
and step S71, if the convergence condition is met, ending the topology optimization process, and obtaining the topology configuration of the embedded movable piezoelectric driving compliant mechanism meeting the minimum size control.
In some embodiments, in step S7, the step of solving the Hamilton-Jacobi equation from the obtained virtual speed variable specifically includes:
step S72, updating design variables by adopting a moving progressive line algorithm;
step S73, solving a Hamilton-Jacobi equation based on the virtual speed variable:
in the formula ,for normal speed +.>Representing the velocity of movement of a point on the boundary of the piezoelectric actuator,representing the normal direction vector of the corresponding point.
In order to further verify the effectiveness of the topology optimization design method of the embedded piezoelectric driving compliant mechanism based on size constraint, the displacement inverting amplifier is taken as an example for explaining the invention.
Displacement inverting amplifier design domain, boundary condition and output end, embedded piezoelectric driver initial positionThe size of which is shown in FIG. 2, the domain size is designedIs->The left end is a fixed boundary, a virtual load acts on the midpoint of the right end of the mechanism, the size of the virtual load is 1, and the midpoint of the right end is an output end. Output spring rate->Is->Allowable volume fraction. The target minimum size is set to 4 units. Since the displacement inverting amplifier has symmetry, the half is removed for design and discretized into +.>And 4 node units on the plane. Elastic modulus of the Main Structure Material ∈>Poisson's ratio->Elastic modulus of piezoelectric actuator material +.>Poisson's ratio->。
Fig. 3 and fig. 4 are schematic diagrams of a displacement inverting amplifier topology and a topology controlled by a minimum size respectively, and it can be known from the diagrams that the topology optimization design method of the embedded piezoelectric driving compliant mechanism based on size constraint can effectively realize the minimum size control, is convenient for actual process manufacturing, and has smooth topology boundary. Fig. 5 is a graph of the iterative process of volume fraction and output displacement for topological optimization under dimensional constraints. Fig. 6 is a basic flow chart of the overall scheme.
In summary, the topology optimization design method of the embedded piezoelectric driving compliant mechanism based on size constraint in the embodiment adopts a point density-level set joint topology description model, represents piezoelectric drivers of any shape by using a level set function, obtains density field distribution of a main structure by adopting an independent point density interpolation method, maximizes output displacement of the compliant mechanism as an objective function, takes the volume fraction of the main structure of the compliant mechanism, the piezoelectric drivers and a design domain as constraint conditions, establishes the topology optimization model of the embedded piezoelectric driving compliant mechanism considering size constraint, solves the topology optimization problem by a moving progressive line algorithm, and is different from the traditional method, and the rigidity of the main structure is influenced by each optimized placement of the piezoelectric drivers under the framework of the point density-level set joint topology description; at the same time, changing the topology of the structure will also affect the next movement of the piezoelectric actuator. The two are mutually influenced and jointly act. The method fundamentally eliminates the defect of step-by-step optimization of the traditional design method, and can simultaneously realize the optimal layout of the piezoelectric driver and the optimal design of the bearing structure. According to the invention, the minimum size of the embedded movable piezoelectric driving compliant mechanism is controlled by adding the size constraint, and the finally obtained target configuration can effectively realize the required minimum size, so that the practical process manufacturing is facilitated.
In the description of the present specification, a description referring to terms "one embodiment," "some embodiments," "examples," "specific examples," or "some examples," etc., means that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the present invention. In this specification, schematic representations of the above terms do not necessarily refer to the same embodiments or examples. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.
The foregoing examples illustrate only a few embodiments of the invention and are described in detail herein without thereby limiting the scope of the invention. It should be noted that various modifications and improvements can be made by those skilled in the art without departing from the spirit of the invention, which falls within the scope of the present invention. Accordingly, the scope of protection of the present invention is to be determined by the appended claims.
Claims (9)
1. The topological optimization design method of the embedded piezoelectric driving compliant mechanism based on the size constraint is characterized by comprising the following steps of:
step S1, defining a design domain, a fixed boundary condition, a punishment coefficient, a unit filtering radius, a virtual load size, an elastic modulus of two materials of a main body structure and an embedded component, a Poisson ratio and volume constraint of the two materials of the main body structure and the embedded component, applying virtual spring stiffness of an output end, setting the shape, the size and the position of a piezoelectric driver, applying voltage, dispersing the design domain into plane 4 node units, and initializing a unit node density design variable;
step S2, interpolating the unit node density design variables by using a rational interpolation model, calculating to obtain the density values of each point in the design domain by using the rational interpolation model, and representing the topological configuration of the main structure;
step S3, independently describing a movable piezoelectric driver by adopting a level set function, dispersing a design domain of the level set function into grids with the same size as the plane 4 node units, and expressing the position, the direction and the speed of the piezoelectric driver in the design domain by using the level set function;
s4, adopting a filtering threshold topology optimization scheme, utilizing the design domain, the filtering design domain and the physical domain as calculation frames, constructing a structure indication function to respectively represent a solid phase and a space phase of the main structure, and referring to two dimension constraints to respectively realize the minimum dimension control of the solid phase and the space phase;
s5, taking the volume fraction and the size of the materials of the main structure and the fact that the piezoelectric driver does not interfere with a design domain as constraint conditions, taking the maximization of the output displacement of the compliant mechanism as an objective function, and establishing a topological optimization design model of the embedded movable piezoelectric driving compliant mechanism based on size constraint;
the topological optimization design model of the embedded movable piezoelectric driving compliant mechanism based on the size constraint in the step S5 is as follows:
wherein ,is a point density design variable for describing the distribution of flexible structural material,/->Is a virtual speed design variable for representing the motion of the piezo actuator,/->Is the +.>Pseudo density of individual cells->Describes the translational and rotational angular speeds of the piezo-actuator,/-> and />Representing the structural pseudo-density design variable and the number of piezo-electric actuators, respectively, +.>For output displacement;
representing the volume of material of the body structure; />Volume of material of non-piezoelectric actuator region, +.>Is the volume of the design domain, +.>Is the volume of the piezo-actuator, +.> and />Respectively representing the size control of the main structure and the empty phase;
s6, solving the output displacement, the non-overlapping control of the piezoelectric driver and the minimum size and volume constraint of the main structure by using the topological optimization design model of the embedded movable piezoelectric driving compliant mechanism based on the size constraint, solving the sensitivity of the output displacement, the non-overlapping control and the volume constraint of the main structure of the objective function mechanism by using a chained derivative method, and solving the sensitivity of the minimum size constraint by using a local gradient derivative method;
s7, updating design variables by using a moving asymptotic algorithm, solving a Hamilton-Jacobi equation by using the obtained virtual speed variables, updating a level set function, and judging whether convergence conditions are met;
if the convergence condition is not satisfied, turning to step S2;
and step S71, if the convergence condition is met, ending the topology optimization process, and obtaining the topology configuration of the embedded movable piezoelectric driving compliant mechanism meeting the minimum size control.
2. The topology optimization design method of the embedded piezoelectric driving compliant mechanism based on the size constraint of claim 1, wherein the step S2 specifically comprises:
step S21, calculating to obtain the density value of each point in the design domain by using the following calculation formula and using a shepherd interpolation model:
in the formula ,for the pseudo-density of the ith cell in the body structure,/or->For shepiad interpolation function, +.>Designing a variable number for the point density for interpolation;
step S22, the Young modulus of each node in the design domain is obtained by adopting the following calculation formula and based on the sheplate interpolation model:
in the formula ,is the Young's modulus of the material of the body structure, < >>Is a penalty coefficient.
3. The topology optimization design method of the embedded piezoelectric driving compliant mechanism based on the size constraint of claim 1, wherein the step S3 specifically comprises:
s31, discretizing a level set design domain into finite element grids with the same size as the density interpolation model;
step S32, describing boundaries of the piezoelectric driver by applying a level set function:
wherein ,representing the inner area of the piezo-electric actuator, < >>Representing the boundaries of the piezoelectric actuator.
4. The topology optimization design method of the embedded piezoelectric driving compliant mechanism based on the size constraint of claim 1, wherein the step S4 specifically comprises:
step S41, defining a design domainFilter design field->And physical Domain->And said design domain, said filtering design domain +.>And the following formula is satisfied between the physical domains:
wherein ,is a unit->Neighborhood set in the filtering domain of (2) ->Is the radius of the linear cap filter, +.>Is a unit->Volume of->Is a unit-> and />Center coordinates of +.> and />A weighting function of the distance between them,/->Control the steepness of the approximate Heaviside function,/->Is a critical value.
5. The method for topological optimization design of an embedded piezoelectric driving compliant mechanism based on size constraint according to claim 4, wherein after step S41, the method further comprises:
step S42, defining inflection regions, namely, an area of 1, an area of 2, and respectively constructing two structural index functions representing a solid phase and a space phase of the main structural material:
wherein, superscript and />Respectively representing a solid phase and an empty phase;
step S43, constructing the following two geometric constraints to achieve minimum size control of the solid and air phases, respectively:
wherein ,for discretizing the collection->The total number of elements in (a) satisfies these two constraints, the value of the filtered field will be greater than the threshold value +_1 at the inflection point +_1>A threshold value +.about.2 at less than +.>。
6. The topology optimization design method of the embedded piezoelectric driving compliant mechanism based on the size constraint of claim 1, wherein the step S6 specifically comprises:
step S61, calculating the sensitivity of the objective function, the volume and the non-overlapping constraint of the compliant mechanism to the design variable by applying an analytical shape sensitivity analysis method;
wherein the finite element form of the objective function is expressed as:the derivative of the objective function to the point density design variable is: />Volume constraint->For->The derivative of the individual point density design variables is: />；
The shape sensitivity of the objective function is:
；
wherein ,the shape derivative representing the virtual velocity perturbation vector of the piezoelectric actuator, the non-overlapping constraint is:
。
7. the method for topological optimization design of an embedded piezoelectric driving compliant mechanism based on size constraint according to claim 6, wherein after step S61, the method further comprises:
step S62, obtaining the derivative of the objective function and the constraint function on the virtual speed design variable through a chain derivation rule:
；
wherein ,is->The partial derivatives of the objective function and the constraint function, respectively.
8. The method for topological optimization design of an embedded piezoelectric driving compliant mechanism based on size constraint according to claim 7, wherein after step S62, the method further comprises:
and step S63, solving the sensitivity of the minimum size constraint by using a local gradient derivative method, filtering the density of the control points by using a density filtering method, and obtaining the density of the filtered control points by using a weighted average method of the densities of adjacent control points.
9. The topological optimization design method for the embedded piezoelectric driving compliant mechanism based on the size constraint of claim 1 is characterized by comprising the following steps: in step S7, the step of solving the Hamilton-Jacobi equation from the obtained virtual speed variable specifically includes:
step S72, updating design variables by adopting a moving progressive line algorithm;
step S73, solving a Hamilton-Jacobi equation based on the virtual speed variable:
in the formula ,for normal speed +.>Representing the speed of movement of a point on the boundary of the piezo-electric actuator, < >>Representing the normal direction vector of the corresponding point.
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