An explicit topology optimization design method for compliant hinges for precise positioning

Abstract

The present invention discloses an explicit topology optimization design method for a compliant hinge for precise positioning, including: establishing a topology optimization design area for a compliant hinge, and defining a design domain and a non-design domain; establishing an underlying finite element analysis framework in the design domain and the non-design domain; using a variable width component with a Bezier curve constrained at both ends as the central skeleton to describe the pseudo-density field of the compliant hinge in the design domain; establishing a compliant hinge explicit topology optimization mathematical model based on a variable width component description with a Bezier curve constrained at both ends as the central skeleton; calculating the sensitivity of the compliant hinge explicit topology optimization mathematical model; using the sensitivity to solve the compliant hinge explicit topology optimization mathematical model through an optimization method, and obtaining the optimal design variables, and extracting the optimal design variables can express the compliant hinge profile. The present invention can optimize the design to obtain a compliant hinge with high rotation accuracy, high rotation flexibility, and easy processing and manufacturing.

Description

Technical Field

[0001] The present invention relates to the technical field of precision equipment such as precision transmission, precision positioning and precision manufacturing, and in particular to an explicit topology optimization design method for a compliant hinge for precision positioning. Background Art

[0002] Compliant mechanisms refer to a type of mechanism that uses its own elastic deformation to transmit input force or displacement. The advantages of compliant mechanisms such as no gap, no lubrication, no assembly required, high precision, and high rigidity have long been discovered by scholars and are widely used in aerospace, mechanical engineering, robotics, medical equipment and other fields. Compliant hinges are the core of compliant precision mechanism transmission of motion. A large number of research institutes have shown that the reasonable design of compliant hinge configurations can effectively improve the motion performance of compliant precision mechanisms. Traditional compliant hinges are mainly designed through shape and size optimization, and are mainly divided into straight circle type, conical section type, wheel type, leaf type, cycloid type and multi-notch type. These studies rely on subjective experience to design the basic configuration, and then optimize its shape and size, etc., which makes it difficult to obtain innovative configurations.

[0003] In recent years, there are technologies such as the Chinese invention patent CN 106096158 A that use implicit topology optimization methods to design compliant hinges. This method breaks away from the design concept of relying on trial and error and experiments that lacks a systematic design theory for compliant hinges. However, implicit topology optimization has several core problems. For example, the topology results of the compliant hinge obtained based on implicit topology optimization must be post-processed to establish a connection with the computer-aided design system (CAD), which destroys the optimality of the compliant hinge design; the characteristic dimensions of implicit topology optimization are difficult to control, affecting the manufacturability of the final result; the calculation amount of implicit topology optimization increases with the increase in the division accuracy of the finite element mesh, which can easily exceed the processing range of the optimization algorithm; and the optimization results are prone to checkerboard phenomenon. Summary of the invention

[0004] In order to solve at least one of the problems existing in the prior art, the present invention proposes an explicit topology optimization design method for flexible hinges for precise positioning, by explicitly optimizing the flexible hinge's variable width component based on a Bezier curve with constrained control points at both ends as the central skeleton, to design a flexible hinge with high rotational accuracy, high rotational flexibility and easy processing and manufacturing.

[0005] To achieve the purpose of the present invention, the present invention provides an explicit topology optimization design method for a compliant hinge for precise positioning, comprising the following steps:

[0006] Step 1: Establish a topology optimization design area for the compliant hinge, define a design domain according to the length-to-width ratio of the compliant hinge, the design domain is a flexible area, and define a non-design domain, the non-design domain is a rigid area;

[0007] Step 2: Establishing the underlying finite element analysis framework of the topology optimization design model of the compliant hinge in the design domain and the non-design domain. Establishing the underlying finite element analysis framework includes discretizing the design domain and the non-design domain into finite grid elements.

[0008] Step 3: In the underlying finite element analysis framework, a variable width component with a Bezier curve constrained at both ends as the central skeleton is used to describe the pseudo-density field of the flexible hinge in the design domain. The variable width component is defined as a Bezier component, wherein by changing The design variables of the Bessel component are used to express the position, size, and geometric shape of the Bessel component in the design domain, and describe the topological configuration of the compliant hinge;

[0009] Step 4: Establish a mathematical model for explicit topology optimization of a compliant hinge based on Bessel component description. The mathematical model for explicit topology optimization of a compliant hinge is:

[0010] ;

[0011] In the formula, find means to find the optimal solution in the definition domain; min means that the solution goal is to minimize the objective function; represents all the design variables, denotes the design variables of all Bessel components, with superscript represents the transpose operation, Indicates The design variables of the Bessel components are , represents the total number of Bezier components, represents the domain of the design variable, represents the objective function, Design domain Any point in and are the structural flexibility of the compliant hinge in the horizontal and vertical axis directions, and They represent the initial flexibility values ​​in the horizontal and vertical axes in the topology optimization iteration, is the weight parameter, represents the volume constraint function, represents the pseudo-density field in the design domain of the current topology, represents the total volume of the design domain, is a given volume ratio constraint value, represents the integral over the domain, represents the global topological description function of all Bessel components in the design domain, is the global stiffness matrix of the current design domain and the non-design domain as a whole, Non-design domain The load applied along the positive direction of the horizontal axis is Non-design domain The load applied along the negative direction of the ordinate axis is and The load and load The displacement field under action;

[0012] Step 5: Calculate the sensitivity of the explicit topology optimization mathematical model of the compliant hinge;

[0013] Step 6: Using the sensitivity, solve the explicit topology optimization mathematical model of the compliant hinge through an optimization method, update the design variables, until the iteration converges, and optimize the optimal design variables. The optimal design variables can be extracted to express the compliant hinge profile, and a new type of compliant hinge design can be obtained.

[0014] Furthermore, in step 1, the design domain and the non-design domain are defined, the specific shape of the design domain is set to a rectangle, the non-design domain is set to a rectangle, and the height of the non-design domain is equal to the height of the design domain.

[0015] The side length ratio of the non-design domain rectangle can be set according to the design space ratio required in practice.

[0016] Furthermore, in step 2, the step of establishing the underlying finite element analysis framework specifically includes: discretizing the design domain and the non-design domain into N and n 4-node square finite grid elements respectively.

[0017] Furthermore, in step 3, by changing The design variables of a variable width component with a Bezier curve constrained at both end control points as the central skeleton express its position, size, and geometry in the design domain, which in turn determines the compliant hinge configuration, that is, the distribution of the material in the design domain.

[0018] Define a variable width component with a Bezier curve constrained at both ends as the central skeleton as a Bezier component, where the central skeleton of the Bezier component is a Bezier curve defined by P control points, and the design variables of each Bezier component include the coordinate variables of 2×P control points that control the central skeleton of the Bezier component and a half-width variable of the Bezier component. According to the difference between constraints and freedom, the design variables of a Bezier component include the following two parts:

[0019] ;

[0020] ;

[0021] ;

[0022] in, Indicates the ordinal number of the Bezier component. , represents the total number of Bezier components, Indicates The Bezier components are constrained and cannot be freely iterated. Indicates The horizontal coordinates of the first and last control points of the central skeleton of a Bezier component, the variable Constrain (fix) the boundaries on both sides of the design domain to ensure the validity of the results. Indicates The Bessel components are not subject to any constraints and can freely iterate the design variables within the domain of the design variables. Indicates The coordinates of the control points of the Bezier curve center skeleton of the Bezier component, the variable It is The half-width variable of the Bezier component, which means Half the width of a Bezier component;

[0023] By defining the design variables of all variable width components with the Bezier curves constrained at both end control points as the central skeleton according to the above method, the position, size and geometric shape of all Bezier components in the design domain can be determined.

[0024] Sure The design domain can be expressed by the design variables of the Bessel components The interior area of ​​a Bezier component And the corresponding contour boundary . In particular, Represents the design domain Zhongyoudi The interior area defined by the Bezier components, Represents the design domain Zhongyoudi The contour boundary defined by the Bezier components. Since the Bezier components are only defined in the design domain So only a single topological description function is constructed for each Bessel component in the design domain , Design domain Any point in the middle;

[0025] Define the zero isosurface of a single topological description function (i.e. = 0) represents the contour boundary interface of the Bezier component. Located inside a Bezier component, >0; when the point Located at the border of a Bezier component, <0; when the point Located outside the Bezier component, =0. Then, the design domain Zhongyoudi The definition expression of a single topological description function corresponding to a Bessel component is:

[0026] ;

[0027] Each Bessel component is solved by the definition expression of a single topological description function to obtain a single topological description function of a single Bessel component ; After obtaining a single topological description function for a single Bessel component, by For each point Calculate 8 single topological description functions The maximum value of can be obtained by taking the global topological description function of all Bessel components, which is expressed as:

[0028] ;

[0029] In the formula, is the global topological description function, The area ≥0 is the design domain The union of the interior area defined by all Bezier components in and the boundary;

[0030] Based on the global topological description function, it can be further transformed into a pseudo-density field. Using the Heaviside function Represents the pseudo density field of the material in the design domain, Heaviside function The expression is:

[0031] ;

[0032] Heaviside Function The global topology description function Converted to a pseudo-density field with a 0-1 distribution, where a function value of 1 represents the presence of material at that point, and a function value of 0 represents the absence of material;

[0033] In order to make the Heaviside function differentiable, use the smoothed Heaviside function ,Will Substitute to get the smoothed pseudo-density field:

[0034] ;

[0035] In the formula, represents the pseudo-density field in the design domain of the current topology, Take a small positive number to avoid the singularity of the stiffness matrix. is a parameter used to control the size of the smoothing interval.

[0036] Furthermore, in step 3, by constraining the coordinates of the control points at the head and tail ends of the central skeleton of the Bezier component, the horizontal coordinates of the control points at the head and tail ends are fixed to the boundaries on both sides of the design domain, and the fixed points are selected as load input points or load output points. This method ensures the existence of material at this position, thereby ensuring the validity of the design results.

[0037] Furthermore, in step 4, the design domain needs to be set and non-design domain Young's modulus and Poisson's ratio of the material.

[0038] Furthermore, in step 4, in order to ensure the versatility of processing and manufacturing, the performance of the two ends of the flexible hinge needs to be the same. You can choose to add symmetry settings about the X-axis and Y-axis to the flexible hinge to achieve a flexible hinge configuration that is symmetrical about the horizontal and vertical directions. The entire design domain is divided into four quadrants based on the X-axis and Y-axis, and one of the quadrants is defined as the symmetry reference domain. , for the remaining three quadrants , , The global topological description function in is symmetrically mapped. At this time, the mathematical model of explicit topological optimization of the compliant hinge is:

[0039] ;

[0040] In the formula, , , Respectively represent quadrants , , The global topological description function of represents the global topological description function of the symmetric parameter quadrant, Represents the coordinates of any point in the corresponding quadrant.

[0041] Preferably, there are four symmetric reference conditions for the uniaxial compliant hinge, that is, four quadrants are respectively used as symmetric reference domains, and the setting method is as follows:

[0042] First, the entire design domain is divided into four quadrants based on the horizontal symmetry axis (X axis) and the vertical symmetry axis (Y axis), and one of the quadrants is selected as the symmetry reference domain. , for example, the upper left quadrant (the symmetric reference domain set in the present invention is not limited to selecting the upper left quadrant).

[0043] Then the global topological description functions in the remaining three quadrants are symmetrically mapped. The specific settings are:

[0044] About The horizontal mirror quadrant of satisfy:

[0045] ;

[0046] About The vertical mirror image quadrant of satisfy:

[0047] ;

[0048] About The central symmetric mirror quadrant of satisfy:

[0049] ;

[0050] Then, the corresponding constraint expression in the mathematical model of explicit topology optimization of flexible hinges is:

[0051] ;

[0052] in, , , Respectively , , The global topological description function of represents the global topological description function of the symmetric reference domain, Represents the coordinates of any point in the corresponding quadrant.

[0053] Furthermore, in step 4, in order to ensure the feasibility of processing and manufacturing, the minimum half width of the Bezier component is limited by a width constraint expression to ensure its manufacturability. The width constraint expression is:

[0054] ;

[0055] In the formula, Indicates the maximum value of the half-width variable of the Bezier component during the iteration process.

[0056] Furthermore, in step 5, the objective function and the volume constraint function are solved for the mathematical model of the explicit topology optimization of the hinge using the adjoint matrix method. A design variable in a Bezier component The partial derivative of , we get:

[0057] The sensitivity of the objective function is:

[0058] ;

[0059] The sensitivity of the volume constraint function is:

[0060] ;

[0061] In the formula, represents the smoothed Heavyisede function about The partial derivatives of a single topological description function for a Bessel component, Indicates A single topological description function for each of the four Bezier components.

[0062] Furthermore, in step 6, the obtained sensitivity is used to solve the mathematical model of explicit topology optimization of the compliant hinge through the MMA algorithm.

[0063] Compared with the prior art, the present invention has at least the following advantages and beneficial effects:

[0064] (1) Without any design experience, the method of the present invention can be used to obtain the topological configuration of the compliant hinge with the best performance under the current objectives and constraints. Compared with the traditional compliant hinge designed based on intuition and experience, the new compliant hinge obtained by the present invention has higher rotational flexibility and precision, and lower profit level.

[0065] (2) The existing topology optimization technology for designing flexible hinges is usually unable to explicitly express the contour boundary of the optimization result, which is easy to destroy the optimality of the result during post-processing, and the problems of checkerboard and too many design variables are easy to appear during the design process. The method proposed in the present invention explicitly describes the topological configuration of the flexible hinge based on the movable Bezier component, which can establish the connection between the topological optimization result and the CAD / CAE software. The topological configuration of the flexible hinge is directly expressed by the coordinates and width values ​​of the central skeleton control points of the Bezier component constrained at both ends. The boundary information of the current optimization result can be obtained by directly extracting the design variables of the Bezier component, without the need for secondary processing of tracing (no need for post-processing tracing); and the number of design variables is small, which is only determined by the configuration of the Bezier component (the topological configuration of the flexible hinge is directly expressed by the coordinates and width information of the central skeleton control points of the Bezier component constrained at both ends, and the design variables include the coordinates and width of the control points of the Bezier component, and there is no need to use the pseudo-density of the grid point as the design variable), which provides a new design idea and method for the design of flexible hinges. BRIEF DESCRIPTION OF THE DRAWINGS

[0066] Figure 1It is a flowchart of the steps of an explicit topology optimization design method for a compliant hinge for precise positioning in an embodiment of the present invention.

[0067] Figure 2 It is a schematic diagram of the design domain and the non-design domain, dimensions, boundary conditions and loads in an embodiment of the present invention.

[0068] Figure 3 It is a schematic diagram of topological description of expressing a compliant hinge using Bezier components constrained at both ends in an embodiment of the present invention.

[0069] Figure 4 It is a schematic diagram of the topology optimization results of an embodiment of the present invention under different upper limits of volume ratio.

[0070] Figure 5 It is a plan view of a novel flexible hinge obtained in an embodiment of the present invention.

[0071] Figure 6 It is a schematic diagram of a three-dimensional model of a novel flexible hinge obtained in an embodiment of the present invention.

[0072] Figure 7 It is a three-dimensional model diagram of the bridge-type amplification mechanism.

[0073] Figure 8 It is a schematic diagram of setting the symmetric reference domain in an embodiment of the present invention. DETAILED DESCRIPTION

[0074] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present application, not all of the embodiments. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of this application.

[0075] Please refer to Figure 2-4 The present invention provides an explicit topology optimization design method for a compliant hinge for precise positioning, comprising the following steps:

[0076] Step 1: Establish a topology optimization design area for the compliant hinge, define a design domain according to the length-to-width ratio of the compliant hinge, and the design domain is a flexible area; define a non-design domain, and the non-design domain is a rigid area.

[0077] The design domain is set according to the design space ratio of actual requirements, and the side length ratio of the non-design domain is set according to the design space ratio of actual requirements. Set as a rectangle, in this embodiment, the side length A square of millimeters is used as the design domain, but the side length ratio of the design domain of the present invention is not limited thereto (for example, the design domain can also be set as a rectangle). Non-design domain (Rigid member) set to long ,Width mm rectangular area, thickness is In other embodiments, the design domain and the non-design domain may also be set to other shapes, but the boundary conditions and the corresponding flexibility need to be changed accordingly.

[0078] Step 2: Establish the underlying finite element analysis framework in the design domain and non-design domain, including discretizing the design domain and non-design domain into finite mesh elements.

[0079] In one embodiment of the present invention, the underlying finite element analysis framework refers to discretizing the design domain into 50×50 4-node square finite grid units, discretizing the non-design domain into 50×150 4-node square finite grid units, and numbering each grid node. Overall, the underlying finite element analysis framework contains 50×200 4-node square finite grid units consisting of 51×201 numbered grid nodes (the number of finite grid units is not limited to this and can be set according to actual applications).

[0080] Step 3: In the underlying finite element analysis framework, a variable width component with a Bezier curve constrained at both ends as the central skeleton is used to describe the pseudo-density field of the flexible hinge in the design domain. The design variables of the variable width component are used to express the position, size, and geometric shape of the variable width component in the design domain, and describe the topological configuration of the flexible hinge.

[0081] In this step, by changing The design variables of a variable width component with a Bezier curve constrained at both ends as the central skeleton express its position, size, and geometric shape in the design domain, thereby determining the configuration of the compliant hinge, that is, the distribution of the material in the design domain. The variable width component with a Bezier curve constrained at both ends as the central skeleton is defined as a Bezier component. In one embodiment of the present invention, 8 Bezier components are used to describe the topological structure of the compliant hinge, and the Bezier curve defined by 8 control points constitutes the central skeleton of the Bezier component (it can be understood that the control points of the central skeleton of the Bezier component are not limited to 8. For the construction requirements of the Bezier curve, the control points are at least 2; the number of control points is set according to the control designer's requirements for the smoothness of the local area. The higher the required local smoothness, the more control points need to be set; the position of the control point directly determines the shape of the Bezier curve as the central skeleton of the Bezier component, thereby determining the position and shape of the Bezier component in the design domain. The size (thickness) of the Bezier component is determined by the width variable). It can be understood that in other embodiments, the number of Bezier components is not limited to 8 and is set according to the complexity of the actual problem.

[0082] In this embodiment, eight Bezier components are used to describe the topological structure of the compliant hinge. The design variables of any Bezier component include the coordinate variables of the 16 control points of the central skeleton of the Bezier component and the half-width variable of one Bezier component. According to the difference between constraint and freedom, the design variables of a Bezier component include the following two parts:

[0083] ;

[0084] ;

[0085] ;

[0086] in, Indicates the ordinal number of the Bezier component. , Indicates The Bezier components are constrained and cannot be freely iterated. Indicates The horizontal coordinates of the first and last control points of the central skeleton of a Bezier component, the variable Constraints (fixed) are placed on both sides of the design domain to ensure the validity of the results. Indicates The Bessel components are not subject to any constraints and can freely iterate the design variables within the domain of the design variables. Indicates The coordinates of the control points of the Bezier curve center skeleton of the Bezier component, the variable Indicates The half-width variable of the Bezier component is Half the width of the Bezier component.

[0087] According to the above method, the design variables of all variable width components with Bezier curves constrained at both end control points as the central skeleton are defined, so that the position, size and geometric shape of all Bezier components in the design domain can be determined.

[0088] Figure 3 The topological description diagram of the flexible hinge expressed by the two-end constrained Bezier components in the embodiment of the present invention, wherein the left figure is the central skeleton configuration of each Bezier component and the boundary contour constructed by the union of each component; the right figure is all Schematic representation of the zero isosurface in the design domain.

[0089] After determining the design variables of the eight Bessel components, the design domain can be expressed as The interior area of ​​a Bezier component And the corresponding contour boundary . In particular, Represents the design domain Zhongyoudi The interior area defined by the Bezier components; Represents the design domain Zhongyoudi The contour boundary is defined by the Bezier components. Since the Bezier components are only defined in the design domain So only in the design domain Construct a single topological description function for each Bezier component , Design domain Any point in the middle.

[0090] In this method, the zero isosurface of a single topological description function (i.e. = 0) to represent the contour boundary interface of the Bezier component. Located inside a Bezier component, >0; when the point Located at the border of a Bezier component, <0; when the point Located outside the Bezier component, = 0. In particular, the design domain Zhongyoudi The definition expression of a single topological description function corresponding to a Bessel component is:

[0091] ;

[0092] The eight Bessel components are solved by the definition expression of the single topological description function to obtain the single topological description of each Bessel component. ; After obtaining a single topological description function for a single Bessel component, by For each point Calculate 8 single topological description functions The maximum value of can obtain the global topological description function of all Bessel components. The expression of the global topological description function is:

[0093] ;

[0094] represents the global topological description function, The area ≥0 is the design domain The union of the interior area defined by all Bezier components in and the bounds.

[0095] Based on the global topological description function, it can be further transformed into the target pseudo-density field. Represents the material pseudo-density field in the design domain, and the Heaviside function expression is:

[0096] ;

[0097] Heaviside Function The global topology description function Convert to a pseudo-density field with a 0-1 distribution, where the function value 1 represents the point If there is material, the function value 0 means there is no material. In order to make the Heaviside function differentiable, the present invention uses the smoothed Heaviside function ,Will Substitute to get the smoothed pseudo-density field:

[0098] ;

[0099] a pseudo-density field within the design domain representing the current topology; Take a small positive number to avoid the singularity of the stiffness matrix; is a parameter used to control the size of the smoothing interval, and is generally a small positive number. In one embodiment of the present invention, Take 0.003, Take 0.6 times the minimum side length of the grid element.

[0100] Step 4: Determine the objective function, design variables, constraints and boundary conditions, and establish a mathematical model for explicit topology optimization of the compliant hinge based on a variable width component (Bezier component) described by a Bezier curve with constrained control points at both ends as the central skeleton.

[0101] After obtaining the pseudo-density field of the material in the design domain in step 3, in this step, first set the design domain and non-design domain Young's modulus and Poisson's ratio of the material; then set the boundary conditions: The load along the positive direction of the horizontal axis and the load along the negative direction of the vertical axis are applied to the midpoint of the right end, and then the displacement field is obtained by using the finite element solver. The finite element calculation formula is:

[0102] ;

[0103] ;

[0104] in is the global stiffness matrix of the current design domain and the non-design domain as a whole, Non-design domain The load applied along the positive direction of the horizontal axis is specifically in the non-design domain The load applied at the midpoint of the right end, Non-design domain The load applied along the negative direction of the ordinate axis is specifically in the non-design domain. The load applied at the midpoint of the right end, and The load and load The displacement field under action.

[0105] definition is the structural flexibility of the compliant hinge in the horizontal axis direction :

[0106] ;

[0107] definition is the structural flexibility of the compliant hinge in the vertical axis direction :

[0108] ;

[0109] An ideal flexible hinge should have greater flexibility in the direction of rotation and greater stiffness (less flexibility) in the direction of undesired motion. Therefore, the objective function of the flexible hinge topology optimization is defined as maximizing the flexibility of the flexible hinge in the ordinate axis direction and minimizing the flexibility of the flexible hinge in the abscissa axis direction. The objective function is:

[0110] ;

[0111] The design variables are:

[0112] ;

[0113] represents all design variables; They represent the design variable groups of 8 variable width components with Bezier curves constrained at both end control points as the central skeleton; represents the domain of the design variable; represents the objective function; and They represent the initial flexibility values ​​in the horizontal and vertical axes in the topology optimization iteration respectively; is a weight parameter, whose domain is [0,1]. In one embodiment of the present invention, The value is 0.5.

[0114] The constraint condition for defining the volume is the ratio of the material in the design domain to the volume of the design domain, expressed as:

[0115] ;

[0116] represents the volume constraint function; a pseudo-density field within the design domain representing the current topology; represents the total volume of the design domain; represents the integral over the domain, A global topological description function representing the global Bessel component in the design domain; is a given volume ratio constraint value, and its value range is generally [0,1]. In one embodiment of the present invention, The value is 0.3.

[0117] Based on the above analysis, the mathematical model of explicit topology optimization of compliant hinges is:

[0118] .

[0119] In one embodiment of the present invention, in order to ensure the universality of processing and manufacturing, the performance of the two ends of the flexible hinge must be the same. Add symmetry settings about the X-axis and the Y-axis to the flexible hinge to achieve a flexible hinge configuration that is symmetrical about the horizontal and vertical directions. First, the entire design domain is divided into four quadrants based on the horizontal axis and the vertical axis, and the upper left quadrant is selected as the symmetry reference domain. (It can be understood that the present invention is not limited to setting the symmetric reference domain in the upper left quadrant, and other quadrants can also be set as symmetric reference domains in other embodiments) to perform symmetric mapping on the global topological description functions in the remaining three quadrants. Figure 8 , the specific settings are:

[0120] Quadrant is about the symmetric reference domain The horizontal mirror quadrant of satisfy:

[0121] ;

[0122] Quadrant is about the symmetric reference domain The vertical mirror image quadrant of satisfy:

[0123] ;

[0124] Quadrant is about the symmetric reference domain The central symmetric mirror quadrant of satisfy:

[0125] ;

[0126] The corresponding constraint expression is:

[0127] ;

[0128] in , , Respectively represent quadrants , , The global topological description function of represents the global topological description function of the symmetric reference domain, Represents the coordinates of any point in the corresponding quadrant.

[0129] Therefore, when there is a requirement for manufacturing versatility, the mathematical model of explicit topology optimization of compliant hinges is:

[0130] ;

[0131] In one embodiment of the present invention, in order to ensure the feasibility of processing and manufacturing, a half-width variable constraint is added to the mathematical model of the explicit topology optimization of the flexible hinge, that is, the minimum half-width of the Bessel component is limited by the following width constraint expression to ensure its manufacturability. The constraint expression is:

[0132] ;

[0133] In the formula, Indicates the maximum value of the half-width variable of the Bezier component during the iteration process.

[0134] Step 5: Calculate the sensitivity of the explicit topology optimization mathematical model of the compliant hinge.

[0135] In this step, for the mathematical model of explicit topology optimization of the flexible hinge with variable width components based on the Bezier curve with constrained control points at both ends as the central skeleton established in step 4, the adjoint matrix method is used to solve the objective function and the volume constraint function with respect to the first A design variable in a Bezier component The partial derivative of , we get:

[0136] The sensitivity of the objective function is:

[0137] ;

[0138] The sensitivity of the volume constraint function is:

[0139] ;

[0140] In the formula, represents the smoothed Heavyisede function about The partial derivatives of a single topological description function for a Bessel component, Indicates A single topological description function for each Bezier component, is the design domain.

[0141] Step 6: Use the sensitivity obtained in step 5 to solve the explicit topology optimization mathematical model of the compliant hinge through the optimization method, update the design variables, and optimize the optimal design variables until the iteration converges. The optimal design variables can be directly extracted to express the compliant hinge contour and obtain a new type of compliant hinge design.

[0142] The final design result of the compliant hinge can be obtained by directly extracting the design variables to express the contour boundary.

[0143] In one embodiment of the present invention, the optimization method adopted is the moving asymptote method (MMA algorithm). It is understandable that other optimization methods can be adopted in other embodiments.

[0144] Figure 4 Shown are different given volume ratio constraint values The topological result diagram of Figure 4It is shown that the optimal result under the current constraints can be obtained by setting different volume ratio constraint values, which proves the effectiveness and versatility of the method of the present invention.

[0145] In one embodiment of the present invention, by setting a given volume ratio constraint value and By optimizing the design, a specific new type of flexible hinge can be obtained using the method of the present invention. Figure 5 FIG. 1 is a schematic plan view of a novel compliant hinge obtained by an explicit topology optimization method in this embodiment. Figure 6 This is a three-dimensional model diagram of the new flexible hinge.

[0146] By Figure 5 and Figure 6 The new compliant hinge and the elliptical compliant hinge shown in the figure are applied to the bridge displacement amplification mechanism with the same size ratio to establish a three-dimensional model. Figure 7 The three-dimensional model diagrams of two bridge-type amplification mechanisms are shown, where (a) is a schematic diagram of a bridge-type displacement amplification mechanism provided with an elliptical compliant hinge, and (b) is a schematic diagram of a bridge-type displacement amplification mechanism provided with the aforementioned novel compliant hinge. The bridge-type displacement amplification mechanism is imported into the commercial software ABAQUS for finite element analysis, and a 40-micron displacement is applied to the displacement input end to calculate the output displacement, displacement amplification ratio, parasitic displacement, and relative parasitic displacement of the two bridge-type displacement amplification mechanisms. The comparison results are shown in Table 1:

[0147] Table 1 Performance comparison of two displacement amplification mechanisms

[0148]

[0149] By comparison, it can be found that compared with the elliptical compliant hinge, the new compliant hinge obtained by the present invention has a larger output displacement and a smaller parasitic displacement when used in a precision transmission mechanism, indicating that the compliant hinge optimized by the method of the present invention has a greater rotational flexibility and a higher rotational accuracy.

[0150] The method provided in the aforementioned embodiment of the present invention adopts an explicit topology optimization method of a variable width component based on a Bezier curve with constrained control points at both ends as the central skeleton to solve the problems of the existing implicit topology optimization that cannot directly extract geometric information, feature size is difficult to control, checkerboard phenomenon and large design variables. Through the present invention, a flexible hinge with high rotation accuracy, high rotation flexibility and easy processing and manufacturing can be optimized.

[0151] The above description of the disclosed embodiments enables those skilled in the art to implement or use the present invention. Various modifications to these embodiments will be apparent to those skilled in the art, and the general principles defined in the present invention may be implemented in other embodiments without departing from the spirit or scope of the present invention. Therefore, the present invention will not be limited to the embodiments shown herein, but will conform to the widest scope consistent with the principles and novel features disclosed in the present invention.

Claims

1. An explicit topology optimization design method for compliant hinges for precise positioning, characterized in that: The following steps are involved: Step 1: Establish a topology optimization design area for a compliant hinge and define a design domain and a non-design domain, wherein the design domain is a flexible area and the non-design domain is a rigid area; Step 2: Establishing an underlying finite element analysis framework in the design domain and the non-design domain, wherein establishing the underlying finite element analysis framework includes discretizing the design domain and the non-design domain into finite grid elements; Step 3: In the underlying finite element analysis framework, a variable width component with a Bezier curve constrained at both ends as the central skeleton is used to describe the pseudo-density field of the flexible hinge in the design domain. The variable width component is defined as a Bezier component, wherein by changing M The design variables of the Bessel component are used to express the position, size, and geometric shape of the Bessel component in the design domain, and describe the topological configuration of the compliant hinge; Step 4: Establish a mathematical model for explicit topology optimization of a compliant hinge based on Bessel component description. The mathematical model for explicit topology optimization of a compliant hinge is: In the formula, find means to find the optimal solution in the definition domain; min means that the solution goal is to minimize the objective function; represents all the design variables, denotes the design variables of all Bessel components, with superscript represents the transpose operation, Indicates The design variables of the Bessel components are , represents the total number of Bezier components, represents the domain of the design variable, represents the objective function, Design domain Any point in and are the structural flexibility of the compliant hinge in the horizontal and vertical axis directions, and They represent the initial flexibility values ​​in the horizontal and vertical axes in the topology optimization iteration, is the weight parameter, represents the volume constraint function, represents the integral over the domain, represents the pseudo-density field in the design domain of the current topology, represents the global topological description function of all Bessel components in the design domain, represents the total volume of the design domain, is a given volume ratio constraint value, is the global stiffness matrix of the current design domain and the non-design domain as a whole, Non-design domain The load applied along the positive direction of the horizontal axis is Non-design domain The load applied along the negative direction of the ordinate axis is and The load and load The displacement field under action; Step 5: Calculate the sensitivity of the explicit topology optimization mathematical model of the compliant hinge; Step 6: Using the sensitivity, solve the explicit topology optimization mathematical model of the compliant hinge through an optimization method, update the design variables, until the iteration converges, optimize the optimal design variables, extract the optimal design variables to express the compliant hinge profile, and complete the compliant hinge design.

2. The method for explicit topology optimization design of a compliant hinge for precise positioning according to claim 1, characterized in that: In step 1, when defining the design domain and the non-design domain, the shape of the design domain is set to a rectangle, the non-design domain is set to a rectangle, and the height of the non-design domain is equal to the height of the design domain.

3. The method for explicit topology optimization design of compliant hinges for precise positioning according to claim 1, characterized in that: In step 4, the design domain is also set and non-design domain Young's modulus and Poisson's ratio of the material.

4. The method for explicit topology optimization design of compliant hinges for precise positioning according to claim 1, characterized in that: In step 3, the central skeleton of the Bezier component is a Bezier curve defined by P control points. The design variables of each Bezier component include the coordinate variables of the 2×P control points that control the central skeleton of the Bezier component and the half-width variables of each Bezier component. According to the difference between constraints and freedoms, the design variables of a Bezier component include: ； ； ； in, Indicates the ordinal number of the Bezier component. , represents the total number of Bezier components, Indicates The Bezier components are constrained and cannot be freely iterated. Indicates The horizontal coordinates of the first and last control points of the central skeleton of a Bezier component, Indicates The Bessel components are not subject to any constraints and can freely iterate the design variables within the domain of the design variables. Indicates The coordinates of the control point of the center skeleton of the Bezier component, the variable Indicates The half-width variables of the Bezier components; By defining the design variables of all Bessel components, the position, size, and geometry of all Bessel components in the design domain can be determined; Sure The design domain can be expressed by the design variables of the Bessel components The interior area of ​​a Bezier component And the corresponding contour boundary , Represents the design domain Zhongyoudi The interior area defined by the Bezier components, Represents the design domain Zhongyoudi The contour boundary defined by the Bezier components; Define the zero isosurface of a single topological description function: = 0 to represent the contour boundary interface of the Bezier component. When inside a Bezier component, >0; when the point Located at the border of a Bezier component, <0; when the point Located outside the Bezier component, =0, then the design domain Middle The definition expression of a single topological description function corresponding to a Bessel component is: ； Each Bessel component can obtain a single topological description function through the definition expression of a single topological description function. After obtaining a single topological description function for each Bessel component, we can use For each point calculate A single topological description function The maximum value of the global topological description function of all Bessel components is obtained, and the expression is: ， In the formula, is the global topological description function, The area ≥0 is the design domain The union of the interior area defined by all Bezier components in and the boundary, Indicates A single topological description function for each Bezier component; Based on the global topological description function, it is converted into a pseudo-density field using the Heaviside function Represents the pseudo density field of the material in the design domain, Heaviside function The expression is: ； Heaviside Function The global topology description function Converted to a pseudo-density field with a 0-1 distribution, where a function value of 1 indicates the presence of material and a function value of 0 indicates the absence of material.

5. The method for explicit topology optimization design of compliant hinges for precise positioning according to claim 4, characterized in that: In order to make the Heaviside function differentiable, use the smoothed Heaviside function ,Will Substitute to get the smoothed pseudo-density field: In the formula, represents the pseudo-density field in the design domain of the current topology, Take a small positive number to avoid the singularity of the stiffness matrix. is a parameter used to control the size of the smoothing interval.

6. The method for explicit topology optimization design of compliant hinges for precise positioning according to claim 4, characterized in that: In step 3, by constraining the coordinates of the control points at the head and tail ends of the central skeleton of the Bezier component, the horizontal coordinates of the control points at the head and tail ends are fixed to the boundaries of the design domain on both sides, and the fixed points are selected as load input points or load output points.

7. The method for explicit topology optimization design of compliant hinges for precise positioning according to claim 4, characterized in that: In step 4, the symmetry settings about the X-axis and the Y-axis are added to the flexible hinge to realize the flexible hinge configuration that is symmetrical about the horizontal and vertical directions. The entire design domain is divided into four quadrants based on the X-axis and the Y-axis, and one of the quadrants is defined as the symmetry reference domain. , for the remaining three quadrants , , The global topological description function in is symmetrically mapped. At this time, the mathematical model of explicit topological optimization of the compliant hinge is: In the formula, , , Respectively represent quadrants , , The global topological description function of represents the global topological description function of the symmetric parameter quadrant, Represents the coordinates of any point in the corresponding quadrant.

8. The method for explicit topology optimization design of compliant hinges for precise positioning according to claim 4, characterized in that: In step 5, the sensitivity is calculated using the adjoint matrix method by solving the objective function and the volume constraint function about the A design variable in a Bezier component The partial derivative of , the sensitivity of the objective function and the sensitivity of the volume constraint function are obtained, and the expression is: The sensitivity of the objective function is: ； The sensitivity of the volume constraint function is: ； In the formula, represents the smoothed Heavyisede function about The partial derivatives of a single topological description function for a Bessel component, Indicates A single topological description function for each Bezier component.

9. The method for explicit topology optimization design of compliant hinges for precise positioning according to claim 1, characterized in that: In step 4, the minimum half-width of the Bezier component is also limited by a width constraint expression to ensure its manufacturability. The width constraint expression is: ， In the formula, Indicates the maximum value of the half-width variable of the Bezier component during the iteration process.

10. An explicit topology optimization design method for a compliant hinge for precise positioning according to any one of claims 1 to 9, characterized in that: In step 6, the sensitivity of step 5 is used to solve the mathematical model of explicit topology optimization of the compliant hinge through a moving asymptote algorithm.

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