Dielectric elastomer shape and multi-material soft robot topological optimization method

Abstract

The invention provides a dielectric elastomer shape and multi-material soft robot topological optimization method, which comprises the following steps of: defining a soft robot structure design domain, initializing design variables and boundary conditions, punishing a stiffness matrix by adopting a sequence multi-material interpolation model, and calculating the shape of a dielectric elastomer by utilizing an Ohm law and a voltage attenuation coefficient model. The electric field intensity of the dielectric elastomer is calculated, a Maxwell stress field of the dielectric elastomer is solved by utilizing a hyperelastic material theoretical model in combination with a thermodynamic framework, and a mechanical equilibrium equation is constructed to map the stress field into an equivalent node load of the structure to solve the displacement response of the structure. Establishing a topological optimization mathematical model of the dielectric elastomer driving shape and the multi-material soft robot; solving the target function and the constrained sensitivity information, and correcting the sensitivity information by using a Heaviside filtering method; and updating a design variable by adopting a moving asymptotic algorithm, and obtaining an optimal ontology configuration when the topological optimization model meets a convergence condition criterion.

Description

Technical Field

[0001] The present invention relates to the technical field of dielectric elastomer drive mechanism optimization, and in particular to a dielectric elastomer shape and multi-material soft robot topology optimization method. Background Art

[0002] Soft robots generally use intelligent materials such as dielectric elastomers (DE), shape memory alloys, shape memory polymers, etc. as drives. Dielectric elastomers have the advantages of large electro-induced deformation, high energy density, fast response, and good output controllability, making soft robots driven by them an important research object.

[0003] Dielectric elastomer mechanisms driven by dielectric elastomers are often covered with compliant electrodes to maximize the drive input, and the output is completed by an external mechanism connected to the dielectric elastomer. The mechanism needs to have a certain strength to support the pre-stretched dielectric elastomer, while being able to deform and output force and displacement. Only by integrating the dielectric elastomer into the mechanism through reasonable design can the required functions be achieved and its driving effect be better exerted.

[0004] Moreover, as the requirements for mechanism performance in engineering practice are increasing, the topological optimization design results of single-phase material mechanisms can no longer meet the requirements for high comprehensive performance of mechanisms. Using multiple materials instead of single-phase materials for mechanism topological optimization design can make full use of the different properties of various materials. By matching multiple materials with different strengths, the requirements for engineering mechanism design can be better met. The topological optimization method is used to design multi-material mechanisms, and the optimal topological configuration of the multi-material mechanism is sought under the constraints, so that a multi-material mechanism with the best performance can be obtained.

[0005] At present, the design of soft robots is mainly based on bionic principles and the creator's intuition. It requires a deep understanding of the material structure and properties to design some soft robots with the required performance, which makes this type of design usually inefficient and challenging. Therefore, it is urgent to develop a theoretical system suitable for the design of soft robots. Summary of the invention

[0006] Based on this, the purpose of the present invention is to provide a dielectric elastomer shape and multi-material soft robot topology optimization method to at least solve the deficiencies in the above-mentioned technology.

[0007] The present invention provides a dielectric elastomer shape and multi-material soft robot topology optimization method, comprising: Step S1: defining the structural design domain, constraint boundary conditions, and initial voltage value of the soft robot, setting the initial value of the design variable, the virtual spring stiffness of the output end, the penalty coefficient, the conductivity coefficient of the electrode area and the non-electrode area, the filter radius, the elastic modulus of the material, the Poisson's ratio, and the volume constraint; Step S2: Penalize the elastic modulus of the material unit using a multi-material sequential difference model, and calculate the overall stiffness matrix of the body structure of the multi-material soft robot; Step S3: Calculate the electric field strength of the dielectric elastomer using Ohm's law and the voltage attenuation coefficient model, solve the Maxwell stress field of the dielectric elastomer by combining the theoretical model of hyperelastic material with the thermodynamic framework, and calculate the equivalent nodal load of the structure in combination with the equilibrium equation to solve the displacement response of the structure; Step S4: Construct a topology optimization model for the dielectric elastomer and the multi-material soft robot with the maximum displacement response as the objective function and the structural volume fraction and strain capacity as the constraints; Step S5: Solve the sensitivities of the objective function with respect to the design variables and the sensitivities of the structural volume and strain energy constraints with respect to the design variables from the topology optimization models of the dielectric elastomer and the multi-material soft robot, and correct the sensitivity information through the Heaviside filtering method; Step S6: Update the design variables using the moving asymptote algorithm, and determine whether the convergence condition of the optimization algorithm is satisfied: If not, re-execute Step S2; If satisfied, the topology optimization process ends, and the optimal driving shape of the dielectric elastomer and the body configuration of the multi-material soft robot are obtained.

[0008] Further, Step S2 includes: Normalize the element density to convert each material parameter into a dimensionless relative value; Penalize the elastic modulus of the material unit using a multi-material sequential interpolation model; Calculate the overall stiffness matrix of the body structure of the multi-material soft robot.

[0009] Further, Step S3 includes: Determine the conductivity coefficient of the electrode region, establish an ordinary differential equation based on the electric field model problem with a voltage attenuation coefficient, and solve the electric field strength distribution according to the set boundary conditions; Establish an equilibrium equation based on the law of conservation of charge and solve it using the Galerkin method to obtain the state equation of the electric field; Derive the constitutive model of the dielectric elastomer electromechanical coupling system through the hyperelastic material theory model combined with the thermodynamic analysis framework, and calculate the Maxwell stress field of the dielectric elastomer.

[0010] Derive the equivalent nodal load of the structure according to the equilibrium equation, thereby obtaining the displacement field of the structure.

[0011] Further, the topology optimization model for the dielectric elastomer and the multi-material soft robot is: ; Among them, is the objective function, is the position vector of the output displacement, is the displacement vector, T represents matrix transpose, is the elastic modulus of the th cell, is the cell density of the th cell, is the overall charge density matrix, is the electric field strength matrix, is the overall stiffness matrix, represents the overall nodal force array, is the Maxwell stress matrix of the dielectric elastomer, is the volume constraint, represents the current volume fraction, represents the current density, represents the allowable volume fraction, is the strain energy constraint, is the current strain energy, is the defined strain energy, represents the density of the cell density of the th cell after density filtering, represents the Heaviside density of the cell density of the th cell after being processed by the

[0012] Further, the step S5 includes: Calculating the sensitivity of the objective function with respect to the design variables by using the Lagrange multiplier method, and deriving the sensitivities of the volume constraint and the strain energy constraint with respect to the design variables; Using the Heaviside filtering method to correct the sensitivity information of the objective function and the constraints.

[0013] In the dielectric elastomer shape and multi-material soft robot topology optimization method of the present invention, during the topology optimization process, high-stiffness materials can be automatically distributed in the support structure to maintain stability, while low-modulus materials are concentrated in the remaining areas to maximize deformation. Compared with single-material designs, the multi-material combination can break through the contradiction between the stiffness and deformation ability of a single material, achieve an increase in output displacement under the same volume constraint, realize the collaborative optimization of dielectric elastomer actuation and structural load-bearing, and provide a better topology configuration design paradigm for high-response, large-deformation soft robots. Description of the Drawings

[0014] Figure 1 Flowchart of the method for topology optimization of the dielectric elastomer shape and multi-material soft robot in the first embodiment of the present invention; Figure 2 Schematic diagram of the design domain of the multi-material soft robot model driven by dielectric elastomers in the first embodiment of the present invention; Figure 3 Dielectric elastomer-driven shape and topology configuration of the multi-material soft robot body in the first embodiment of the present invention; Figure 4 Dielectric elastomer-driven shape topology configuration in the first embodiment of the present invention; Figure 5 Topology configuration of the multi-material soft robot body in the first embodiment of the present invention.

[0015] The following specific embodiments will further illustrate the present invention in conjunction with the above drawings. Specific Embodiment

[0016] To facilitate the understanding of the present invention, the present invention will be described more comprehensively below with reference to the relevant drawings. Several embodiments of the present invention are given in the drawings. However, the present invention can be implemented in many different forms and is not limited to the embodiments described herein. On the contrary, these embodiments are provided to make the disclosure of the present invention more thorough and comprehensive.

[0017] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by those skilled in the technical field to which the present invention belongs. The terms used herein in the specification of the present invention are only for the purpose of describing specific embodiments and are not intended to limit the present invention. The term "and / or" used herein includes any and all combinations of one or more of the related listed items.

[0018] Embodiment 1 Please refer to Figure 1 , which shows the method for topology optimization of the dielectric elastomer shape and multi-material soft robot in the first embodiment of the present invention. The method specifically includes steps S101 to S106: S101, define the structure design domain, constraint boundary conditions, and initial voltage value of the soft robot, set the initial values of the design variables, virtual spring stiffness at the output end, penalty coefficient, electrode area, conductivity coefficients of the non-electrode area, filter radius, elastic modulus, Poisson's ratio, and volume constraint of the material; Please refer to Figure 2 , which shows the schematic diagram of the design domain of the soft robot in this embodiment.

[0019] S102. Use a multi - material sequence difference model to penalize the elastic modulus of material elements, and calculate the overall stiffness matrix of the body structure of the multi - material soft robot; In specific implementation, normalize the element density so that each material parameter is converted into a dimensionless relative value:

[0020] In the formula, and are the normalized density and the actual density of material respectively, is the maximum value of the density among all candidate multi - materials, is the number of multi - materials.

[0021] Use a multi - material sequence interpolation model to penalize the elastic modulus of material elements:

[0022] In the formula, and are the elastic modulus and density of the th element respectively, is the penalty coefficient.

[0023] Among them, the calculation formulas for the proportionality coefficient and the translation coefficient are: ; ; In the formula, among them, and are the solid elastic modulus and density of material respectively, and represent the normalized elastic modulus and density of material respectively, .

[0024] Furthermore, calculate the overall stiffness matrix of the body structure of the multi - material soft robot.

[0025] S103. Use Ohm's law and the voltage attenuation coefficient model to calculate the electric field strength of the dielectric elastomer, use the theoretical model of hyperelastic material combined with the thermodynamic framework to solve the Maxwell stress field of the dielectric elastomer, and combine the equilibrium equation to calculate the equivalent nodal load of the structure to solve the displacement response of the structure; In specific implementation, the conductivity coefficient of the electrode region is determined, an ordinary differential equation is established based on the electric field model problem with a voltage attenuation coefficient, and the electric field strength distribution is solved according to the set boundary conditions; the ordinary differential equation of the electric field model problem with a voltage attenuation coefficient can be expressed as: ; In the formula, is the electric potential, is the electric field, is the conductivity matrix, is the voltage attenuation coefficient, is the distance from the power supply.

[0026] Among them, the boundary conditions are: ; In the formula, is the electric potential, is the input voltage.

[0027] Specifically, substituting the boundary conditions to obtain the electric field strength distribution: ; In the formula, , is the conductivity coefficient of the non-electrode part, is the ratio of the input voltage to the voltage at the distance from the power supply, that is .

[0028] Based on the law of conservation of charge, an equilibrium equation is established and solved using the Galerkin method to obtain the state equation of the electric field. According to the law of conservation of charge: ; In the formula, and respectively represent two directions, is the voltage loss, is the voltage attenuation coefficient, , are respectively , the electric current amounts in the directions, is the conductivity, is the total current change amount.

[0029] Substituting Ohm's law, , the Laplace equation of the electric potential can be obtained: ; To avoid directly dealing with the second derivative, the Galerkin method is used to convert the differential equation into an integral weak form. Select the trial function , and convert the differential equation into an integral equation: ; Among them, each is constructed by using the same basis function, and N represents the total number of elements. The design domain is discretized into finite elements, and each element defines a shape function matrix: ; Among them, is the basis function of node i.

[0030] The electric potential inside the element is interpolated from the nodal electric potential as follows: ; In the formula, the nodal electric potential , and T represents the matrix transpose.

[0031] Similarly, the interpolation expression of the trial function can be obtained: ; In the formula, is the th trial function of the element; Using integration by parts and Green's theorem to solve the integral equation: ; In the formula, is the boundary normal of the surface , and here becomes . Substituting the weak form into the shape function, we get: ; In the formula, the gradient matrix , represents the shape function, represents the external reference electric potential, is the global transformation matrix. Therefore, the state equation of the electric field can be obtained: ; Among them, is the global charge density matrix, is the electric field strength matrix, is the global load vector.

[0032] Derive the constitutive model of the electro - dielectric elastomer electromechanical coupling system through the hyperelastic material theory model combined with the thermodynamic analysis framework, and calculate the Maxwell stress field of the electro - dielectric elastomer. The second - order tensor form of the stress of the electro - dielectric elastomer under the action of electromechanical coupling can be expressed as: ; Among them,i, j represents the coordinate direction of the current configuration (after deformation), K is the coordinate direction of the reference configuration (undeformed), represents the direction in the material coordinate system K mapped to the direction in the spatial coordinate system j , represents the direction in the material coordinate system K mapped to the direction in the spatial coordinate system i , is the deformation gradient described in [reference], which is a second-order tensor representing the mapping relationship before and after material deformation; det is the determinant operator, representing the calculation of the F determinant of the deformation gradient, is the vacuum permittivity; is the expression of the strain energy density function of the material, which can be written in the form of the deformation gradient; , , are the components of the electric field strength in the i, j, k direction respectively; is the Kronecker symbol. The expression of the total stress is divided into two terms. The first term is the elastic force part, and the second term is the tensor expression of the Maxwell stress.

[0033] According to the equilibrium equation, the equivalent nodal loads of the structure are derived to obtain the displacement field of the structure. Among them, the force equilibrium equation of the infinitesimal volume element: ; In the formula, , , are the body force components in the , , directions respectively. is the stress. In the plane case, represents the thickness , and can be rewritten as .

[0034] In the discrete design, . Usually, in the finite element design, the external basic forces generated by the body force and the traction can be expressed as: ; In the formula, , where is the identity matrix. is the physical array, and the traction force is not considered in this embodiment , assume , the above formula can be simplified to: ; In the global form, the nodal load can be calculated using the global transformation matrix and the global stress vector , that is: ; In the formula, the global transformation matrix can be obtained by assembling the element transformation matrix .

[0035] S104. Taking the maximum displacement response as the objective function and the structural volume fraction and strain capacity as the constraints, a topology optimization model of the dielectric elastomer and the multi-material soft robot is constructed; In specific implementation, taking the maximum output displacement as the objective function and the structural volume fraction and strain energy as the constraints, a topology optimization design model of the dielectric elastomer shape and the multi-material soft robot is established. The topology optimization model of the dielectric elastomer shape and the multi-material soft robot is: ; Among them, is the objective function, is the position vector of the output displacement, is the displacement vector, T represents matrix transpose, is the elastic modulus of the th element, is the element density of the th element, is the overall charge density matrix, is the electric field strength matrix, is the overall stiffness matrix, represents the overall nodal force array, is the global transformation matrix, is the Maxwell stress matrix of the dielectric elastomer, is the volume constraint, represents the current volume fraction, represents the current density, represents the allowable volume fraction, is the strain energy constraint, is the current strain energy, is the defined strain energy, represents the density of the th element after density filtering of the element density, represents the The density of each unit after being processed by Heaviside the filtering algorithm, represents the number of units.

[0036] S105. Solve the sensitivities of the objective function with respect to the design variables, and the sensitivities of the structural volume and strain energy constraints with respect to the design variables by the topological optimization model of the dielectric elastomer and the multi-material soft robot, and correct the sensitivity information through the Heaviside filtering method; Furthermore, the step S105 specifically includes steps S1051 to S1052: S1051. Calculate the sensitivity of the objective function with respect to the design variables by using the Lagrange multiplier method, and deduce the sensitivities of the volume constraint and the strain energy constraint with respect to the design variables; S1052. Correct the sensitivity information of the objective function and the constraints by using the Heaviside filtering method.

[0037] In specific implementation, solve the sensitivities of the objective function with respect to the design variables, and the sensitivities of the structural volume and strain energy constraints with respect to the design variables by the shape of the dielectric elastomer and the topological optimization model of the multi-material soft robot, and correct the sensitivity information through the Heaviside filtering method.

[0038] Among them, the sensitivity of the objective function to the design variables is: ; In the formula, is the objective function, is the density after density filtering, is the stiffness matrix, is the inverse matrix of the stiffness matrix, is the density after heaviside filtering, is the displacement column array, is the global transformation matrix, is the Maxwell stress matrix, is the overall charge density matrix, is the inverse matrix of the overall charge density matrix, is the electric field strength matrix.

[0039] The sensitivity of the volume constraint to the design variables is: ; In the formula, is the volume constraint, is the current volume fraction, N is the total number of units; The sensitivity of the strain energy constraint to the design variables is: ; In the formula, is the strain energy constraint, is the given strain energy; S106. Update the design variables by using the moving asymptote algorithm, and judge whether the convergence condition of the optimization algorithm is satisfied: If not satisfied, re - execute step S102; If satisfied, the topology optimization process ends, and the optimal dielectric elastomer driving shape and the multi - material soft robot body configuration are obtained.

[0040] To further verify the effectiveness of the dielectric elastomer shape and multi - material soft robot topology optimization method in this embodiment, this embodiment is explained by numerical examples of the dielectric elastomer shape and multi - material soft robot topology optimization. The numerical examples are all two - dimensional structures.

[0041] The semi - symmetric design domain, boundary conditions and input voltage of the dielectric elastomer shape and multi - material soft robot topology optimization model are as Figure 3 shown. An initial voltage is applied at the left end of the design domain. To facilitate placing an object, a vacant area with a size of is set at the lower - right edge. L x and L y represent the lengths in the x and y directions of the design domain. There is also a solid non - design domain with a size of near the upper - left corner. The design domain is discretized into 160×80 four - node elements, the minimum filtering radius is , the allowable volume fraction is , and the spring stiffness at the output end is N / m to imitate the stiffness of the clamped object; the set Neo - Hookean model parameters are: shear modulus C10 = 16000, incompressibility coefficient D1 = 1×10^10, the elastic modulus of the material is set to 1 GPa, and the stretch ratio is 3.2 times. The material parameters are shown in Table 1.

[0042] Table 1 Material parameter table of the gripper

[0043] Figure 3 is the dielectric elastomer driving shape and the multi - material soft robot body topology configuration. Among them, the light - gray part is the dielectric elastomer driving shape, and the black and dark - gray parts are the multi - material soft robot body configuration. Its output displacement is 1.8714 mm, which is caused by Figure 4 and Figure 5The effectiveness of the optimal dielectric elastomer shape and multi-material soft robot topology optimization obtained in this embodiment is verified.

[0044] In summary, for the dielectric elastomer shape and multi-material soft robot topology optimization method in the above embodiments of the present invention, during the topology optimization process, high-stiffness materials can be automatically distributed in the support structure to maintain stability, while low-modulus materials are concentrated in the remaining areas to maximize deformation. Compared with the single-material design, the multi-material combination can break through the contradiction between the stiffness and deformation ability of a single material, achieve an increase in output displacement under the same volume constraint, realize the collaborative optimization of dielectric elastomer actuation and structural load-bearing, and provide a better topology configuration design paradigm for high-response and large-deformation soft robots.

[0045] The technical features of the above embodiments can be combined arbitrarily. For the sake of brevity of description, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, it should be considered as the scope recorded in this specification.

[0046] The above embodiments only represent several implementation manners of the present application. The description is relatively specific and detailed, but it should not be construed as a limitation on the scope of the invention patent. It should be noted that for those of ordinary skill in the art, without departing from the concept of the present application, several modifications and improvements can still be made, and these all belong to the protection scope of the present application. Therefore, the protection scope of the patent of the present application should be subject to the appended claims.

Claims

1. A topology optimization method for the shape of dielectric elastomers and multi-material soft robots, characterized in that, Including: Step S1: Define the structural design domain of the soft robot, the constraint boundary conditions, the initial voltage value, set the initial values of the design variables, the virtual spring stiffness at the output end, the penalty coefficient, the electrode area, and the conductivity coefficients of the non-electrode area, the filtering radius, the elastic modulus of the material, the Poisson's ratio, and the volume constraint; Step S2: Use the multi-material sequential interpolation model to penalize the elastic modulus of the material elements and calculate the overall stiffness matrix of the body structure of the multi-material soft robot; Step S3: Use Ohm's law and the voltage attenuation coefficient model to calculate the electric field strength of the dielectric elastomer, use the theoretical model of the hyperelastic material combined with the thermodynamic framework to solve the Maxwell stress field of the dielectric elastomer, and combine the equilibrium equation to calculate the equivalent nodal load of the structure to solve the displacement response of the structure; Step S4: Construct the topology optimization model of the dielectric elastomer and the multi-material soft robot with the maximum displacement response as the objective function and the structural volume fraction and strain capacity as the constraints; Step S5: Solve the sensitivity of the objective function with respect to the design variables and the sensitivity of the structural volume and strain energy constraints with respect to the design variables from the topology optimization model of the dielectric elastomer and the multi-material soft robot, and correct the sensitivity information by the Heaviside filtering method; Step S6: Use the moving asymptote algorithm to update the design variables and judge whether the convergence condition of the optimization algorithm is satisfied: If not satisfied, re-execute Step S2; If satisfied, the topology optimization process ends, and the optimal driving shape of the dielectric elastomer and the body configuration of the multi-material soft robot are obtained.

2. The dielectric elastomer shape and multi-material soft robot topology optimization method according to claim 1, characterized in that The said Step S2 includes: Normalize the element density to convert each material parameter into a dimensionless relative value; Use the multi-material sequential interpolation model to penalize the elastic modulus of the material elements; Calculate the overall stiffness matrix of the body structure of the multi-material soft robot.

3. The dielectric elastomer shape and multi-material soft robot topology optimization method according to claim 1, wherein The said Step S3 includes: Determine the conductivity coefficient of the electrode area, establish an ordinary differential equation based on the electric field model problem with a voltage attenuation coefficient, and solve the electric field strength distribution according to the set boundary conditions; Establish an equilibrium equation based on the law of conservation of charge and solve it using the Galerkin method to obtain the state equation of the electric field; Derive the constitutive model of the dielectric elastomer electromechanical coupling system through the theoretical model of the hyperelastic material combined with the thermodynamic analysis framework and calculate the Maxwell stress field of the dielectric elastomer; According to the equilibrium equation, derive the equivalent nodal load of the structure, thereby obtaining the displacement field of the structure.

4. The dielectric elastomer shape and multi-material soft robot topology optimization method according to claim 1, characterized in that The topology optimization model of the dielectric elastomer and the multi-material soft robot is: ； Among them, is the objective function, is the position vector of the output displacement, is the displacement vector, T represents matrix transpose, is the elastic modulus of the th element, is the element density of the th element, is the overall charge density matrix, is the electric field strength matrix, is the overall stiffness matrix, represents the overall nodal force column matrix, is the global transformation matrix, is the Maxwell stress matrix of the dielectric elastomer, is the volume constraint, represents the current volume fraction, represents the current density, represents the allowable volume fraction, is the strain energy constraint, is the current strain energy, is the defined strain energy, represents the density of the th element after density filtering of the element density, represents the density of the th element after being processed by the Heaviside filtering algorithm, represents the number of elements.

5. The dielectric elastomer shape and multi-material soft robot topology optimization method according to claim 1, characterized in that The said Step S5 includes: Use the Lagrange multiplier method to calculate the sensitivity of the objective function with respect to the design variables, and derive the sensitivity of the volume constraint and the strain energy constraint with respect to the design variables; Use the Heaviside filtering method to correct the sensitivity information of the objective function and the constraints.

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