A flexible sensor design method and device based on topology optimization

Abstract

The present invention belongs to the technical field related to the design of flexible electronic sensor structures. It discloses a flexible sensor design method and device based on topology optimization, comprising the following steps: (1) assigning a design variable to each grid of the design domain of the flexible sensor to be designed based on variable density topology optimization, thereby obtaining a physical density field representing the structural parameters; (2) calculating the resistance of the flexible sensor to be designed after large deformation, and combining parallel calculation to obtain all resistance values ​​under finite stretching conditions; (3) calculating the sensitivity of the resistance to the design variables under all conditions by using analytical sensitivity derived based on the adjoint method; (4) constructing a topology optimization model with the goal of minimizing the error between the calculated resistance and the target resistance-strain response, and solving the topology optimization model based on the moving progressive MMA algorithm to achieve continuous updating and optimization of the design variables to obtain the final optimization result. The present invention can realize programmable large deformation resistance.

Description

Technical Field

[0001] The present invention belongs to the technical field related to the design of flexible electronic sensor structures, and more specifically, relates to a flexible sensor design method and equipment based on topology optimization. Background Art

[0002] Flexible electronics is an emerging electronic technology with broad application prospects in aerospace, healthcare, and communications. Compared to traditional silicon-based electronics, flexible electronics offer unparalleled advantages, such as deformability and ultra-thinness. These advantages have become a hotly pursued academic frontier for research institutions and scholars both domestically and internationally.

[0003] Flexible strain sensors, fabricated from flexible, stretchable materials and conductive metals, have widespread applications in health monitoring and smart industry. Because metals inherently lack stretchability, stretchable sensing structures based on metals are required to generate a resistance-strain response. Furthermore, the sensor's sensitivity decreases with increasing strain, leading to increased measurement errors. Therefore, designing a stretchable sensing structure whose resistance varies linearly with strain is crucial.

[0004] Micro-nano sensing structures, as two-dimensional functional structures with a width of microns and a thickness of nanometers, are ubiquitous in flexible electronic systems. In order to adapt to large deformation service environments, micro-nano sensing structures need to have both deformability and stable electrical properties, which has become a bottleneck in the design of flexible electronics. The "ultra-thinness" of flexible electronics enables them to have out-of-plane bending capabilities, so the main breakthrough in deformability is in-plane stretchability. In addition, the resistance of the micro-nano sensing structure of the strain sensor needs to change linearly with strain, which requires a design method that can obtain the ability to have arbitrary resistance-strain response, which can also be called programmable resistance-strain response. Therefore, the design method of a resistance-programmable stretchable sensing structure system is one of the key technical issues in the design of flexible electronic sensing structures.

[0005] Solid metals, such as silver, copper, and gold, have good electrical conductivity and stable mechanical and chemical properties, making them ideal materials for conduction and sensing. However, most solid metals are not stretchable, and their electrical properties are unstable, and their resistance will drift uncontrollably with deformation. Currently, existing technologies can already design stretchable metal sensing structures, such as wavy, arc-shaped, serpentine, spiral and other winding structure design technologies, as well as paper-cutting technology. However, so far, there is no structural design technology that can achieve precise control of the resistance response of the sensing structure during stretching (i.e., programmable resistance). Summary of the Invention

[0006] To address the aforementioned deficiencies or improvements in the prior art, the present invention provides a flexible sensor design method and device based on topology optimization. This method leverages the advantages of topology optimization, such as its large design space, lack of initial configuration requirements, and rationality, to precisely control the resistance response of the sensing structure during stretching. Furthermore, the present invention provides an analytical sensitivity expression for the resistance to the structural topology design variables, enabling rapid and accurate calculation of the resistance sensitivity to these design variables. With the goal of minimizing the error between the calculated resistance and the target resistance-strain response, and considering mass and connectivity constraints, a topology optimization model and solution method for a programmable resistance-stretchable sensing structure are established.

[0007] To achieve the above objectives, according to one aspect of the present invention, a flexible sensor design method based on topology optimization is provided, the design method comprising the following steps:

[0008] (1) Based on variable density topology optimization, a design variable is assigned to each grid in the design domain of the flexible sensor to be designed, and the design variables are filtered and mapped to obtain a physical density field representing the structural parameters;

[0009] (2) The resistance of the flexible sensor to be designed after large deformation is calculated by electromechanical coupling finite element analysis under large deformation geometric nonlinearity and material nonlinearity, and all resistance values ​​under finite stretch conditions are obtained by combining parallel calculation;

[0010] (3) Calculate the sensitivity of the resistance to the design variables under all working conditions by using the analytical sensitivity derived based on the adjoint method to obtain the optimized descent direction;

[0011] (4) With the goal of minimizing the error between the calculated resistance and the target resistance-strain response, a topology optimization model is constructed under the conditions of satisfying the structural mass constraint and the structural connectivity constraint. The topology optimization model is solved based on the moving progressive MMA algorithm to achieve continuous updating and optimization of the design variables, and the final optimization result is obtained, that is, the design of the flexible sensor is completed.

[0012] Furthermore, the expression of the topology optimization model is:

[0013]

[0014] str m (u) = f ext -f int (u)=0

[0015] r e (u,v)=K e (u)v=0

[0016]

[0017]

[0018] 0≤ρ i ≤1

[0019] Where N d is the number of intermediate points taken; R i is the actual resistance at the midpoint; ρ is the design variable; k i is the target response slope; ε i is the tensile strain at the midpoint. R0 is the target resistance; f ext is the external node load vector; f int (u) is the internal node load vector; K e (u) is the stiffness matrix of the electrical equation; N e is the number of elements in the design domain; is the structural pseudo-density field; v e is the volume of each unit; is the upper limit of volume fraction; ρ i is the design variable of the i-th unit; v is the potential field distribution of the deformed structure after the potential difference is applied.

[0020] Furthermore, the governing equations for the electromechanical coupling finite element analysis under large deformation geometric nonlinearity and material nonlinearity are:

[0021]

[0022] Where r is the residual node load vector, f ext and f int are the external and internal node load vectors, respectively; u and v are the displacement field and electric potential field after structural deformation, respectively; F and κ are the deformation gradient and electrical conduction matrix, respectively.

[0023] Furthermore, the displacement distribution of the structure after tensile deformation is first solved by geometric nonlinear finite element method, thereby obtaining the deformed structure; then, a voltage difference is applied to the resistance calculation end point on the deformed structure, so as to solve the voltage distribution in the structure by finite element method, and the voltage gradient is calculated to obtain the current value at the end point; finally, the resistance is obtained by the ratio of the voltage difference and the current value at the end point.

[0024] Furthermore, the resistance of the structure is determined by the ratio of the potential difference to the inlet or outlet current, and the corresponding calculation formula is:

[0025]

[0026] Where, v inlet is the potential at the point where the current enters; v outlet The potential of the current source point; is the inlet current value; Γinletis the entrance boundary.

[0027] Furthermore, the calculation formula of analytical sensitivity is:

[0028]

[0029] The adjoint vectors λ1 and λ2 in the formula are obtained from the following governing equations:

[0030]

[0031] in, and Calculated in steps (1)-(2), It can be directly obtained from the relationship between resistance and electric field; The calculation formula is:

[0032]

[0033] Furthermore, when the target in the topology optimization model is optimized to the minimum value 0, R i =(1+k i ε i )R0 holds, and we can get:

[0034]

[0035] Furthermore, given a target resistance R0, a resistance R is obtained after each optimization iteration. i , by calculating R0 and R i The minimum value of the sum of squares of the differences is obtained and iterated continuously until the target in the topology optimization model is optimized to the minimum value 0, that is, R i =(1+k i ε i )R0 is established.

[0036] Furthermore, the design variables are subjected to density filtering and Heaviside mapping to obtain the physical density field representing the structural parameters.

[0037] According to another aspect of the present invention, a flexible sensor design device based on topology optimization is provided, the device including a memory and a processor, the memory storing a computer program, and the processor executing the flexible sensor design method based on topology optimization as described above when executing the computer program.

[0038] In general, compared with the prior art, the flexible sensor design method and device based on topology optimization provided by the present invention have the following beneficial effects:

[0039] 1. This invention is more rational and intelligent than existing technologies. This invention breaks through the empirical limitations of existing technologies and is a rational, intelligent, and computer-based sensor structure design method. It also proposes a topology optimization model for sensor structures with programmable large deformation resistance. Because it eliminates the need for an initial configuration, design experience, and offers a wide design space, it is easier to design innovative configurations with superior, even disruptive, performance.

[0040] 2. The present invention shortens the design process and reduces costs. The present invention integrates computer-aided design (CAD) and computer-aided engineering (CAE) in the design process, breaking through the time-consuming and labor-intensive limitations of the existing technology of manual structural design optimization from CAD to CAE, and can achieve rapid design of sensor structures.

[0041] 3. The present invention has better systematization. The present invention is a system design technology for a sensor structure with programmable resistance-strain response, which can realize innovative sensor structure designs with different resistance-strain responses by setting different parameters.

[0042] 4. This invention offers enhanced scalability, enabling the design of sensing structures with diverse functions by modifying objective functions and constraints. Building upon existing technologies, it can introduce additional thermal and electromagnetic performance constraints to achieve innovative sensing structures that meet multidisciplinary functional requirements, including thermal, mechanical, electrical, and magnetic. Furthermore, because this invention enables the design of arbitrary resistance-strain responses, it can also enable the design of other micro-nano conductive structures besides sensing structures, such as interconnects whose resistance remains constant with strain. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 1 is a flow chart of a flexible sensor design method based on topology optimization provided by the present invention;

[0044] Figure 2 is a schematic diagram of a design domain model provided by an embodiment of the present invention;

[0045] Figure 3 (a) and (b) are the design results corresponding to the different slope resistance changes during the stretching process of the sensing structure of the embodiment of the present invention;

[0046] Figure 4 (a) and (b) are the design results corresponding to the unchanged resistance and nonlinear change of the sensing structure during the stretching process of the embodiment of the present invention, respectively;

[0047] Figure 5 This is the process flow of preparing the flexible sensor according to the designed sensing structure of the present invention. DETAILED DESCRIPTION

[0048] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely for the purpose of explaining the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.

[0049] See also Figure 1 The present invention provides a flexible sensor design method based on topology optimization, which mainly includes the following steps:

[0050] In the first step, a design variable is assigned to each grid of the design domain of the flexible sensor to be designed based on variable density topology optimization, and the design variables are filtered and mapped to obtain the physical density field representing the structural parameters.

[0051] Specifically, the structural design domain is divided into finite element meshes based on actual design requirements, and each mesh is assigned a design variable based on variable-density topology optimization. The design variables are filtered and mapped to obtain a physical density field representing the structural parameters. This physical density field, when a value of 1 or 0 at a cell indicates the presence or absence of material, respectively, enables a topological description of the structure.

[0052] In this embodiment, the design domain and boundary conditions of the structure are determined (such as Figure 2 ), and divide it into grids, setting a cell density ρ for each cell e (e=1,…,N e ), where N e The number of elements in the design domain is calculated by density filtering and Heaviside mapping on the design variables to obtain the physical density field representing the structural parameters.

[0053] Step 2: Calculate the resistance of the flexible sensor to be designed after large deformation through electromechanical coupling finite element analysis under large deformation geometric nonlinearity and material nonlinearity, and combine parallel calculation to obtain all resistance values ​​under finite stretching conditions.

[0054] After a conductive structure undergoes a significant tensile deformation, its resistance changes significantly. Calculating the resistance of a deformed structure using finite element methods involves two steps: first, using geometrically nonlinear finite element methods to determine the displacement distribution of the structure after tensile deformation, thereby obtaining the deformed structure. Then, a voltage difference is applied to the resistance calculation endpoints on the deformed structure. Using finite element methods, the voltage distribution in the structure is determined, and the voltage gradient is calculated to obtain the current value at the endpoints. Finally, the resistance is calculated as the ratio of the voltage difference at the endpoints to the current value.

[0055] In this implementation, the resistance of the structure after large deformation is calculated using a geometric / material nonlinear electromechanical coupling analysis. This step involves electromechanical coupling and consists of two steps: First, the structure is stretched to a given elongation through geometric nonlinear analysis to obtain the displacement field u after the structure is deformed. The governing equation for the displacement solution is:

[0056] r m (u) = f ext -f int (u)=0 (1)

[0057] Where r is the residual node load vector, f ext and f int are the external and internal node load vectors respectively. This embodiment involves the design of a large deformation sensing structure, so a geometric nonlinear model should be used to model the system. The nonlinear strain can be written as:

[0058]

[0059] In the above formula, F is the deformation gradient matrix, which can be calculated by the following formula:

[0060]

[0061] Where X and x represent the coordinate vectors of the initial undeformed structure and the deformed structure, respectively. In this embodiment, the structure is analyzed using a finite element technique based on the total Lagrangian scheme. The Newton Raphson method is used to solve the above nonlinear system, and its corresponding incremental equation is as follows:

[0062] K m Δu=r (4)

[0063] Taking material nonlinearity into account and solving the above equations, the displacement field u can be continuously updated by u + Δu until convergence. Finite element schemes and other nonlinear equation solving methods based on the current structure can also be applied to solve the above equations.

[0064] A voltage difference is applied to the resistance calculation endpoints of the deformed structure, and the electric potential field distribution dependent on the displacement field is solved based on the electromechanical coupling finite element method. The governing equation is:

[0065]

[0066] Where κ is the electrical conductivity matrix. In the initial structure as the reference system, the weak integral form of the above control equation is:

[0067]

[0068] Based on the finite element method, the above integral equation is solved using the following discretized equation:

[0069]

[0070] By solving the above equations, the potential field distribution v of the deformed structure after the potential difference is applied can be obtained.

[0071] The resistance value of the structure after deformation is calculated based on the electric potential field distribution results. The current distribution in the structure can be calculated based on the electric potential field distribution:

[0072]

[0073] The resistance of a structure can be determined by the ratio of the potential difference to the current at the inlet or outlet, i.e.:

[0074]

[0075] Combined with parallel calculation, all resistance values ​​under finite stretch conditions can be quickly obtained.

[0076] Through the above process, the resistance-strain response relationship of a structure can be obtained.

[0077] In step three, the sensitivity of the resistance to the design variables under all working conditions is calculated by the analytical sensitivity derived based on the adjoint method to obtain the optimized descent direction.

[0078] Based on the displacement field, potential field and other information obtained in the above steps, the sensitivity of the resistor to the design variables is calculated. The expression of the analytical sensitivity derived by the adjoint method in this embodiment is:

[0079]

[0080] The above analytical expressions include the derivatives of the resistance, mechanical, and electrical control equations with respect to the design variables and two adjoint vectors λ1 and λ2. The two adjoint vectors can be obtained from the following two control equations:

[0081]

[0082] In the above formula and They have been calculated in the above steps. It can be directly obtained from the relationship between resistance and electric field, leaving the last item The calculation formula is:

[0083]

[0084] By calculating the analytical sensitivity of the above resistance to the design variables, the sensitivity of the resistance to the design variables under all working conditions can be quickly calculated to obtain the descent direction of subsequent optimization.

[0085] In step 4, with the goal of minimizing the error between the calculated resistance and the target resistance-strain response, a topology optimization model is constructed under the conditions of satisfying the structural mass constraints and structural connectivity constraints. The topology optimization model is solved based on the moving progressive MMA algorithm to achieve continuous updating and optimization of the design variables, and the final optimization result is obtained, that is, the design of the flexible sensor is completed.

[0086] The expression of the topology optimization model is:

[0087]

[0088] Where N d is the number of intermediate points taken; R i is the actual resistance at the midpoint; ρ is the design variable; k i is the target response slope; ε i is the tensile strain at the midpoint. R0 is the target resistance; f ext is the external node load vector; f int (u) is the internal node load vector; K e (u) is the stiffness matrix of the electrical equation; N e is the number of elements in the design domain; is the structural pseudo-density field; v e is the volume of each unit; is the upper limit of volume fraction; ρ i is the design variable of the i-th unit; v is the potential field distribution of the deformed structure after the potential difference is applied.

[0089] When the target in the above optimization model is optimized to the minimum value 0, R i =(1+k i ε i )R0 holds, the relationship can be rewritten as:

[0090]

[0091] The optimization model is solved based on the moving progressive MMA algorithm, and the above process is repeated and the design variables are updated to obtain the final optimization result. Given a target resistance R0, a resistance R is obtained after each optimization iteration. i , by calculating R0 and R i The minimum value of the sum of squares of the differences is obtained and iterated continuously until the target in the optimization model is optimized to the minimum value 0, that is, R i =(1+k i ε i )R0 is established.

[0092] After completing the design of the above flexible sensor, the topology optimization structure pattern design can be carried out through photolithography, and the coating and magnetron sputtering thin film deposition technology can be used to prepare a flexible strain sensor with a programmable resistance structure. Figure 5 , specifically including the following steps:

[0093] First, prepare a clean support substrate such as a glass slide or silicon wafer. Spin-coat a layer of PMMA on the support substrate as a sacrificial layer with a thickness of about 1 μm.

[0094] A polyimide film is prepared on a PMMA sacrificial layer. A polyimide precursor solution is spin-coated on the sacrificial layer, pre-cured on a hot plate, and then placed in an oven for imidization to obtain a polyimide film as a film substrate.

[0095] A layer of AZ5214 photoresist (approximately 3 μm thick) was spin-coated onto the polyimide film and pre-cured on a hot plate for 5 minutes. After pre-curing, the mask and support substrate were aligned and exposed for 8.5 seconds. After exposure, the mask was transferred to a developer solution for 50 seconds, replicating the topology-optimized pattern on the mask onto the photoresist.

[0096] A conductive metal film is deposited on the photoresist layer using methods such as evaporation or magnetron sputtering. The thickness of the deposited film is controlled by controlling the deposition rate and time. The supporting substrate is placed in acetone to remove excess photoresist, thereby obtaining a topologically patterned sensing structure. Finally, the flexible sensor is directly peeled off from the substrate using sharp tweezers.

[0097] The present invention obtains a programmable resistance sensing structure that meets the requirements through a completely rational, that is, calculation method. In addition, by setting different k values, sensing structures with different resistance-strain response relationships can be obtained. Figure 3 and Figure 4 Shown are several topology optimization results for different target resistance-strain response relationships. Therefore, this invention represents a systematic and rational approach to designing flexible electronic sensing structures. Compared to existing technologies, this approach offers advantages such as a larger design space and a more rational, intelligent, and systematic design process.

[0098] In this embodiment, the conductive topology is patterned based on photolithography, and a flexible strain sensor is fabricated using thin-film deposition techniques such as magnetron sputtering. Small deformation refers to the deformation of a component due to an external force that is significantly smaller than its original size. Large deformation refers to the large deformation, and the changes in the object's configuration at different times cannot be ignored. During the stretching process, the object's configuration also changes. In this embodiment, the strain reaches 20%. Under such stretching conditions, the metal's shape undergoes significant changes, necessitating the use of large deformation theory for model construction.

[0099] This paper proposes a topology optimization model for a large-deformation programmable resistor sensing structure: by specifying the desired strain magnitude and target stretch resistance, the desired sensing structure can be obtained through topology optimization. Furthermore, within this topology optimization model, the adjoint method is used to determine the analytical sensitivity of the resistor to the design variables (this determines the direction of iteration).

[0100] The present invention also provides a flexible sensor design device based on topology optimization, the device including a memory and a processor, the memory storing a computer program, and the processor executing the flexible sensor design method based on topology optimization as described above when executing the computer program.

[0101] It will be easily understood by those skilled in the art that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A flexible sensor design method based on topology optimization, characterized in that: The method comprises the following steps: (1) Based on variable density topology optimization, a design variable is assigned to each grid in the design domain of the flexible sensor to be designed, and the design variables are filtered and mapped to obtain a physical density field representing the structural parameters; (2) The resistance of the flexible sensor to be designed after large deformation is calculated by electromechanical coupling finite element analysis under large deformation geometric nonlinearity and material nonlinearity, and all resistance values ​​under finite stretch conditions are obtained by combining parallel calculation; (3) Calculate the sensitivity of the resistance to the design variables under all working conditions by using the analytical sensitivity derived based on the adjoint method to obtain the optimized descent direction; (4) With the goal of minimizing the error between the calculated resistance and the target resistance-strain response, a topology optimization model is constructed under the conditions of satisfying the structural mass constraint and the structural connectivity constraint. The topology optimization model is solved based on the moving progressive MMA algorithm to achieve continuous updating and optimization of the design variables, and the final optimization result is obtained, that is, the design of the flexible sensor is completed; The expression of the topology optimization model is: s.t.r m (u)=f ext -f int (u)=0 r e (u,v)=K e (u)v=0 0≤ρ i ≤1 Where N d is the number of intermediate points taken; R i is the actual resistance at the midpoint; ρ is the design variable; k i is the target response slope; ε i is the tensile strain at the midpoint; R0 is the target resistance; f ext is the external node load vector; f int (u) is the internal node load vector; K e (u) is the global stiffness matrix of the electrical equation; Ne is the number of elements in the design domain; is the structural pseudo-density field; v e is the volume of each unit; is the upper limit of volume fraction; ρ i is the design variable of the i-th unit; v is the potential field distribution of the deformed structure after the potential difference is applied; The governing equations for the electromechanical coupling finite element analysis under large deformation geometric nonlinearity and material nonlinearity are: Where r is the residual node load vector, f ext and f int are the external and internal node load vectors, respectively; u and v are the displacement field and electric potential field after structural deformation, respectively; F and κ are the deformation gradient and electrical conduction matrix, respectively.

2. The flexible sensor design method based on topology optimization according to claim 1, characterized in that: First, the displacement distribution of the structure after tensile deformation is solved using geometrically nonlinear finite elements to obtain the deformed structure. Then, a voltage difference is applied to the resistance calculation endpoints on the deformed structure, so that the voltage distribution in the structure is solved using finite elements, and the voltage gradient is calculated to obtain the current value at the endpoint. Finally, the resistance is calculated as the ratio of the voltage difference and the current value at the endpoint.

3. The flexible sensor design method based on topology optimization according to claim 2, characterized in that: The resistance of the structure is determined by the ratio of the potential difference to the inlet or outlet current, and the corresponding calculation formula is: Where, v inlet is the potential at the point where the current enters; v outlet The potential of the current source point; is the inlet current value; Γ inlet is the entrance boundary.

4. The flexible sensor design method based on topology optimization according to claim 1, wherein: The calculation formula of analytical sensitivity is: The adjoint vectors λ1 and λ2 in the formula are obtained from the following governing equations: in, and Calculated in steps (1)-(2), It can be directly obtained from the relationship between resistance and electric field; The calculation formula is:

5. The flexible sensor design method based on topology optimization according to claim 1, wherein: When the target in the topology optimization model is optimized to the minimum value 0, R i =(1+k i ε i )R0 holds, and we can get:

6. The flexible sensor design method based on topology optimization according to claim 5, characterized in that: Given a target resistance R0, a resistance R is obtained after each optimization iteration. i , by calculating R0 and R i The minimum value of the sum of squares of the differences is obtained and iterated continuously until the target in the topology optimization model is optimized to the minimum value 0, that is, R i =(1+k i ε i )R0 is established.

7. The flexible sensor design method based on topology optimization according to any one of claims 1 to 6, characterized in that: Perform density filtering and Heaviside mapping on the design variables to obtain the physical density field representing the structural parameters 8. A flexible sensor design device based on topology optimization, characterized by: The device includes a memory and a processor, the memory stores a computer program, and the processor executes the flexible sensor design method based on topology optimization according to any one of claims 1 to 7 when executing the computer program.

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