CN111090942B - High-sensitivity piezoresistive uniaxial force sensor design method based on topology optimization - Google Patents
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Abstract
The invention discloses a high-sensitivity piezoresistive uniaxial force sensor design method based on topological optimization, which takes the minimum structure external force as an optimization target, combines three constraint conditions of a force balance equation, a volume constraint and a sensitivity constraint, determines the configuration of a sensor by a series of design variables, and for the optimization problem, the invention is based on a mobile component (hole) topological optimization method, unifies the size optimization and the topological optimization design of the sensor by using a topological description function, only needs to add proper relevance to some design variables (geometric control parameters), then maps geometric information in a design domain to a finite element model to obtain an optimized column type finite element discrete format, and solves the problem by using an optimization solver. The method of the invention is expected to reduce the cost of sensor research and development and improve the efficiency of sensor design, and the designed sensor configuration is processable.
Description
Technical Field
The invention relates to automatic design of a sensor, in particular to a high-sensitivity piezoresistive uniaxial force sensor design method based on topology optimization, namely, a piezoresistive uniaxial force sensor meeting the sensitivity requirement is automatically designed by adopting a mobile assembly (hole) topology optimization method.
Background
The force sensor is widely applied to experiments and engineering tests and used for accurately acquiring the force borne by a test piece. With the continuous development of scientific technology, the requirements of the industry on the performance of force sensors are increasing, wherein sensitivity is an important index for evaluating the sensors. Conventional design methods are generally based on engineering experience, and based on the engineering experience, a great deal of time and money are required to be invested for testing and modifying so as to achieve the expected target.
Kang et al and Li et al improve sensor performance by optimizing geometric parameters using shape optimization, but such design methods still require the human pre-specification of the approximate structure of the sensor. Rubio et al and Takezawa et al introduce a topology optimization approach into the sensor design. Based on the solid isopic material with optimization method, they have obtained some sensor configurations that meet the needs. Due to the complex design, most of these sensors cannot be manufactured. Therefore, the technical problem to be solved by the invention is to provide a sensor topology optimization design method capable of being directly processed and produced.
Disclosure of Invention
The invention provides a high-sensitivity piezoresistive single-axis force sensor design method based on topology optimization, which can be used for designing a processable sensor configuration, accelerating the research and development speed of a sensor and saving the research and development cost, aiming at the problems that the traditional sensor design is mostly based on experience, a large amount of trial and error is needed, the research and development cost is high, the sensor configuration given by the traditional automatic design method is not easy to process and produce, and the like.
Generally, the output of a piezoresistive force sensor is realized by attaching a strain gauge at a specific position and converting a force signal into a voltage signal. Then high sensitivity means that the average strain value of the patch area needs to meet the requirements.
The sensor configuration is assumed to be composed of a series of design variables DiIf so, the optimized formula of the present invention can be expressed as：
Wherein C represents the external work of the structure, and a smaller value indicates a stiffer structure. The first equality constraint is the equivalent integral weak form of the force balance equation, u is the true displacement of the sensor structure, and v is any displacement field that satisfies the boundary conditions. Wherein f, t represent the volume force and the surface force applied to the structure,
is the fourth order elastic tensor, and ε is the structural strain corresponding to displacement.
Is a given displacement boundary condition. The second constraint is a volume constraint, meaning that the solid part volume V (D) cannot be larger than the design domain volume
Is determined by the specific ratio f. The requirement for sensor sensitivity is a third constraint. It shows the average strain value ε of each patch areaigThe sum of squares of (a) is required to meet specific requirements. Where n represents the number of strain gages that are connected into the wheatstone bridge. In general, we would like a positive strain value to be added to the bridge, so we use the square to represent this expectation (using an absolute value would result in an irreducible value of 0). Lower limit value of (A) hereinε)2Represents the lowest sensitivity that we expect, and the upper limit value
Which can be interpreted as the maximum allowable strain of the structure obtained by a certain strength theory.
The solving method of the optimization formula mainly comprises the following steps:
1. selection of design variables and topology description functions
Introducing a topology description function phi (x) to represent the distribution of the entity materials in the design domain, wherein the topology description function phi (x) comprises the following steps:
based on the moving assembly (hole) topology optimization method proposed by (Guo 2014; Zhang et al.2016), the configuration of the whole sensor can be determined by the distribution relationship in space for k members. If each member can use the corresponding topology description function phi respectivelyi(x) To describe, then the topological description function of the sensor configuration can be expressed as:
the topological description function of two common geometric holes is introduced
a. Two-dimensional ellipse (Picture 1)
φi(x,y)＝(x'-xic)2/ri1 2+(y'-yic)2/ri2 2-1 (4)
b. Two dimension pole (Picture 2)
φi(x,y)＝(x'-xic)6/ri1 6+(y'-yic)6/ri2(x',η)6-1 (5)
Uniform cross section: r isi2(x', η) is constant
Wherein
The conversion relation between a coordinate system along the main axis of the component and a Cartesian system is shown, and the specific significance of each parameter is shown in figures 1 and 2. R in formula (5)i2(x', η) represents the distribution of the rod thickness along the axial direction, where η is a series of parameters that control this distribution. Based on the selection of the topological description function, the shape and spatial distribution relation of each component will be completely determined by the geometric control parametersAnd (4) determining. And is therefore ranked for the optimization problem described above. These geometric control parameters can be chosen as design variables, namely:
Di＝{xi0,yi0,θi,ri1,ri2(orη)}T (7)
2. finite element formula based on topological description function and ersatz material model
By using an ersatz material model
We can map the geometric information in the design domain onto the finite element model. Wherein the cell density ρe∈[0,1]The solid material fraction of the finite element is shown. RhoeBy 1 is meant that the units are all solid materials and vice versa. EeIs the elastic modulus of the unit, EsIs the modulus of a solid material.
As a step function of regularization
nen then represents the number of nodes of the selected element in the finite element model.
Based on such a mapping relationship, the equilibrium equation of the structure (constraint one) can be expressed as:
K(D)U(D)＝F (10)
wherein D ═ { D ═ D1 T,…Dk TK, U and F represent a rigidity matrix, a displacement vector and a force vector of a finite element equation respectively.
The square of the average strain of the patch area can be expressed as:
wherein
Is the area of the patch region, θigThe patch angle of the strain gauge, Q (theta)ig) Is a rotation matrix. Based on the finite element equation, the integral equation can be expressed using a gaussian integral:
wherein T isθ＝[cos2θ sin2θ 2sinθcosθ]Is Q (theta) in finite element formatig) In the form of a vector.
It is the strain value at the gaussian integration point of ip number in ie number unit. XiipIs the coefficient of the gaussian integral. And JieThe volume change ratio of the standard iso-parametermapping to ie number cell is shown. n iseThe number of cells occupied by the strain gauge # ig, n _ gauge is the number of gaussian integration points selected within the cell.
For the design domain of the rectangular uniform grid, if a four-node Gaussian integral format is adopted, the formula (12) can be simplified to
From the above, we can obtain a finite element discrete format of the optimized equation:
3. optimization solution and sensitivity analysis
For constrained optimization problems, we can use various optimization solvers to solve, such as sequential quadratic programming, moving evolutionary method, and so on.
Taking the moving asymptotic approach as an example, the gradient-like algorithm needs to give derivative values of the objective function and the constraint on the design variables (sensitivity analysis). The sensitivity of the objective function and constraints obtainable using the adjoint method is:
wherein the vector λ can be determined by a concomitant finite element analysis. The adjoint finite element can be expressed as solving an equation on the basis that the original displacement boundary condition is unchanged:
Kλ＝-P (16)
wherein the vector
Is an equivalent force vector derived from sensitivity analysis. Where B isipThe displacement strain mapping matrix on the Gaussian point with the ip number in the element with the ie number is shown. GieAnd the assembly matrix is assembled for unit displacement to integral displacement.
In equation (15), the derivative of the total stiffness to the design variable a can be expressed as:
wherein the derivative of the step function to the design variable may be found by a central difference.
The invention has the beneficial effects that:
1. the method for automatically designing the high-sensitivity piezoresistive single-axis force sensor is expected to reduce the research and development cost of the sensor and improve the design efficiency of the sensor;
2. compared with the existing automatic design method, the geometric information of the method is explicit, which means that the obtained design configuration is machinable;
3. based on the topology description function, the design method can unify the size optimization and the topology optimization design of the sensor. Only some design variables (geometric control parameters) need to be added with proper correlation, some classical sensor configurations, such as S-shaped sensors, can be directly given, and the remaining mutually independent geometric control parameters can be used as design variables for optimizing the sensor shape of the known configuration. The unified expression framework is expected to provide a foundation for further forming a set of sensor automatic design software.
Drawings
FIG. 1 is a geometric sense diagram of geometric control parameters of a two-dimensional elliptical topology description function.
FIG. 2 is a geometric significance diagram of geometric control parameters of a two-dimensional bar topology description function.
FIG. 3 is a flow chart of a high-sensitivity piezoresistive uniaxial force sensor design method based on mobile component method topology optimization.
FIG. 4 is a schematic view of a model "bone-type" single axis pressure sensor, wherein a) geometric information of a model "bone-type" pressure sensor; b) finite element model and geometric details of a certain type of bone type pressure sensor.
Fig. 5 shows an automated design configuration a) of the "bone type" and an iterative process b) of the individual variables.
Detailed Description
The effects of the present invention will be further described with reference to specific examples.
The method of the invention is adopted to carry out secondary design on a certain type of bone type uniaxial pressure sensor. The geometric information and finite element model of the sensor are shown in figure 4, the positions of the patches are positioned on the upper surface and the lower surface of the position x-40, and each patch is attached with a strain gauge to form a half bridge. The calculated strain output value of the model is as follows: epsilong 26.76 e-6. As a comparative example, here we consider only the lower limit of the one-sided strain constraint (S) ((S))ε)2The set value is 7.0 e-6. The topological description function of the bone type is represented by selecting 4 two-dimensional ellipses and 1 uniform-section two-dimensional rod. Only the geometric control parameters of the left lower ellipse and the height of the equal-section rod are used as design variables, and other geometric control parameters are derived through symmetry. The configuration after secondary design and the iteration process of each parameter are shown in figure 5. Table 1 shows the automated designComparison of some parameters of the configuration and the existing configuration. It can be seen that on the basis of slightly improving the strain output (sensitivity) of the patch area, various structural bearing capacity indexes of the automatic design configuration are superior to those of the existing structure. This represents a superior performance of the automated design architecture.
TABLE 1
Claims (1)
1. A high-sensitivity piezoresistive single-axis force sensor design method based on topology optimization is characterized in that the method is used for constructing an optimization model and solving the optimization model as follows:
Find D＝{Di…Dn},u(x)
s.t.
wherein sensor configuration D is defined by a series of
Design variable D within rangeiDetermining that C represents the external force work of the structure, and the smaller the value of C is, the stiffer the structure is;
the first equality constraint is the equivalent integral weak form of the force balance equation, ΩsDesign domain omega for sensor architecturedOccupied area, and has corresponding boundary
u is the true displacement of the sensor structure and v is the set of arbitrary displacement field functions that satisfy the boundary conditions
Any displacement field in the set, where f, t represents the volume and surface forces experienced by the structure,
is the fourth order elastic tensor, epsilon is the structural strain corresponding to the displacement,
is at the boundary of a given displacement boundary condition
An upper displacement value;
the second constraint is a volume constraint, meaning that the solid part volume V (D) cannot be larger than the design domain volume
A specific ratio f;
the third constraint is the sensitivity constraint on the sensor, representing the average strain value ε of the ig patch areaigWhere n represents the number of strain gages connected into the wheatstone bridge, the lower limit value: (ε)2Indicating the desired minimum sensitivity, and an upper limit
Maximum allowable strain for the theoretically obtained structure;
the following method is adopted for solving:
1) selection of design variables and topology description functions
Introducing a topology description function phi (x) to represent the distribution of the entity materials in the design domain, wherein the topology description function phi (x) comprises the following steps:
based on the mobile component topology optimization method, the configuration of the whole sensor can be determined by the distribution relation of k components in space, if each component uses the corresponding topology description function phi respectivelyi(x) To describe, then the topological description function of the sensor configuration can be expressed as:
φ(x)＝max(φi…φk) Method of moving components
φ(x)＝min(φi…φk) Moving the hole method
Based on the selected topological description function, the shape and spatial distribution relation of each component are completely determined by the geometric control parameters in the function;
2) finite element formula based on topological description function and ersatz material model
By using an ersatz material model
Mapping the geometric information in the design domain to a finite element model;
wherein the cell density ρe∈[0,1]Solid material fraction, ρ, representing finite elemente1 means that the units are all solid materials and vice versa; eeIs the elastic modulus of the unit, EsIs the modulus of the solid material;
for the regularized step function:
nen represents the number of nodes of the selected unit in the finite element model;
based on the above mapping relationship, the structural equilibrium equation can be expressed as:
K(D)U(D)＝F
wherein D ═ { D ═ D1 T,…Dk TK, U and F respectively represent a rigidity matrix, a displacement vector and a force vector of a finite element equation;
the square of the average strain of the patch area can be expressed as:
wherein
Is the area of the patch region, θigThe patch angle of the strain gauge, Q (theta)ig) Is a rotation matrix;
based on the finite element equation, the integral equation can be expressed using a gaussian integral:
wherein T isθ＝[cos2θ sin2θ 2sinθcosθ]Is Q (theta) in finite element formatig) In the form of a vector of (a),
the strain value, xi, on the Gauss integral point of ip number in ie number unitipIs the coefficient of Gaussian integral, JieRepresenting the volume change ratio of standard equal-parameter mapping to ie number unit; n iseThe number of the cells occupied by the strain gauge No. ig, and n _ guass is the number of Gaussian integration points selected in the cells;
for the design domain of the rectangular uniform grid, if a four-node gaussian integral format is adopted, the above formula can be simplified as follows:
obtaining a finite element discrete format of an optimized column:
FindD＝{Di…Dn},U(x)
MinimizeC＝FTU(D)
s.t.
K(D)U(D)＝F
Biprepresenting a displacement strain mapping matrix on an ip-number Gaussian point in an ie-number unit;
3) solving the optimization problem with the constraints obtained in the step 2) by adopting an optimization solver.
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