Piezoresistive Sensor Design Using Topology Optimization - CiteSeerX

6th World Congress on Structural and Multidisciplinary Optimization Rio de Janeiro, 30 May - 03 June 2005, Brazil

Piezoresistive Sensor Design Using Topology Optimization Em´ılio Carlos Nelli Silvaa , Bernardo Reis Dreyer de Souzaa , Mauro Gomes Ornelasa and Shinji Nishiwakib a Department of Mechatronics and Mechanical Systems Engineering Escola Polit´ecnica da Universidade de S˜ ao Paulo Av. Prof. Mello Moraes, 2231, S˜ ao Paulo - SP - 05508-900, Brazil b Department of Precision Engineering Graduate School of Engineering Kyoto University Sakyo-ku - Kyoto - 606-8501 - Japan

1. Abstract Piezoresistive materials, materials whose resistivity properties change when subjected to a mechanical stresses, currently have wide industry application for building MEMS, such as, pressure sensors, accelerometers, inclinometers, and load cells. A basic piezoresistive sensor consists of piezoresistive material bonded to a flexible structure, such as a cantilever, membrane, or compliant mechanism, where the flexible structure transfers pressure, force, or inertial force (due to acceleration), thereby causing a stress that changes the resistivity of the piezoresistive material. By applying a voltage to the material, its resistivity can be measured and correlated with the degree of applied pressure or force. The performance of the piezoresistive sensor is closely related to the design of its flexible structure which can be achieved by applying systematic design methods, such as topology optimization. Thus, in this work, a topology optimization formulation has been applied to the design of piezoresistive sensors. As an initial problem, a piezoresistive force sensor design is considered. The optimization problem is posed as the design of a flexible structure that bonded to the piezoresistive material generates the maximum response in terms of resistivity change (or output voltage) when a force is applied.. 2. Keywords:Piezoresistive sensor, topology optimization, design, finite element . 3. Introduction Piezoresistive materials are materials whose resistivity properties change when the material is subjected to mechanical stresses[1]. They are widely applied in industry nowadays to build Microelectromechanical Systems (MEMS) such as pressure sensors[2], accelerometers[3], inclinometers[4], and load cells[5]. A piezoresistive transducer is essentially made of a piezoresistive material bonded to a flexible structure, such as a cantilever, a membrane, or a compliant mechanism. The flexible structure converts a pressure, force, or inertial force (due to acceleration) into strain and thus stress that changes the piezoresistive material resistivity. By applying a voltage to the piezoresistive material, its resistivity can be measured and its value can be related to the applied pressure or force value. Usually the piezoresistive elements are connected in a “Wheatstone bridge” to increase the sensitivity and to compensate temperature effects[1]. Although we can find many works about piezoresistive transducer manufacturing[6], there are few works about piezoresistive transducer modeling and design. Most part of works related to this issue, essentially apply analytical techniques [7, 8] and FEM modelling [4, 9, 10]. Actually there are few and quite recent works [2, 3, 11, 12] where the FEM modelling for piezoresistive materials is discussed. Regarding the use of systematic optimization methods to design these transducers most part of works found are related to the piezoresistive load cell design. Essentially, a parametric optimization is applied to find optimum load cell dimensions to reduce the coupling problem among the strain-gage responses which is a classical problem in load cell design [5]. Thus, in this work, considering the widely application of piezoresistive sensors nowadays, a topology optimization formulation was developed to design piezoresistive transducers. Topology optimization is a computational design method that combines optimization algorithms and finite element method (FEM) to find the optimum topology of mechanical parts considering a desired objective function and some constraints. Recently, topology optimization has been applied to design bio-probe piezoresistive sensors [13], however, in this work, the aim is to develop a more comprehensive topology optimization

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formulation for piezoresistive transducer designs. Piezoresistive transducers can be understood as a compliant mechanism [14] that converts the input force into local output displacement, and so strain and stress, in a set of piezoresistive elements which convert it into output voltage due to the resistivity change. This output voltage can be related to the applied load by using the piezoresistive effect. A classical problem in the piezoresistive transducer design is where to bond or position the piezoresistive material to obtain the highest response[2]. However, the problem can also be posed as the design of a flexible structure for a certain fixed position of the piezoresistive material, that is, the objective is to design an optimum flexible structure that bonded to the piezoresistive material generates the maximum response in terms of resistivity change (or output voltage). To illustrate the method we will focus on the force sensor design. The paper is organized as follows. Section 4 describes the FEM formulation for piezoresistive materials. In Section 5 and 6, the topology optimization formulation for piezoresistive transducer design and its numerical implementation is discussed. In Section 7, some results are presented to illustrate the method. Finally, in Section 8, some conclusions are given. 4. Piezoresistive Fem Formulation 4.1. Piezoresistive Constitutive Equations In piezoresistive materials, resistivity properties are related to mechanical stresses through a constitutive law. Thus, a change in the mechanical stress causes a change in the material resistivity in according to Ohm‘s law[1]: {E} = [ρ] {J} ; {E} = − ▽ φ (1) where φ is the electric potential, {E} and {J} are the electric field strength and current density vectors, respectively, and [ρ] is the anisotropic resistivity matrix given by [1, 2]   ρxx ρxy ρxz [ρ] =  ρxy ρyy ρyz  ⇒ [k] = [ρ]−1 (2) ρxz ρyz ρzz where [k] is the anisotropic conductivity matrix which is the inverse of the resistivity coefficient matrix. In the piezoresistive effect, the resistivity matrix values depend on the material stresses values in according to the following constitutive equation [2]:        σx  1  π11 π12 π12 0 0 0 ρxx                            σy  1  π12 π11 π12 0 0 0  ρyy                             σz 1 π12 π12 π12 0 0 0  ρzz  = + ρ0 = 0  0 0 π44 0 0  σyz  ρyz         0                  0 0 0 π44 0     ρxz     0    0  σxz                σxy 0 0 0 0 0 0 π44 ρxy =

ρ0 {{Im } + [Π] {σ}} = ρ0 {{Im } + [Π] [D] {ε}}

(3)

since {σ} = [D] {ε}, where {σ} is the stress vector, [Π] is the piezoresistive coefficient matrix, [D] is the elasticity coefficient matrix which is orthotropic in the case of piezoresistive materials, and {ε} is the strain vector. Thus, once the mechanical stress is related to the mechanical displacements and the electric field is related to the electric potential, by measuring the electric potential change, the resistivity change, and thus, the displacement change can be obtained. Notice that an electric potential must be applied, so an electrical potential change can be measured. In addition, the piezoresistive effect is not reversible, which means that it can be applied only for sensor purposes and not actuator purposes. In this work, only two-dimensional (2D) problems have been considered. Thus, in the case of a 2D formulation, Eq. (3) becomes:        π11 π12 0  ρxx   1   σx  ρyy = 1 +  π12 π11 0  σy ρ0 =       ρxy 0 0 0 π44 σxy =

ρ0 {{Im } + [Π] {σ}} = ρ0 {{Im } + [Π] [D] {ε}} 2

(4)

4.2. Finite Element Formulation A general method such as the Finite Element Method (FEM) is necessary for the structural analysis since structure with complex topologies are expected. In this work, since only 2D problems are considered, four-node bilinear isoparametric elements are applied based on a plane stress assumption. Consider the piezoresistive sensor medium described in figure 1 with corresponding electrical and mechanical boundary conditions. By applying the discretized finite element (FE) formulation, displacements are approximated using shape functions, and the relation between strain and displacement nodes in each finite element is given by[15] {ε}e = [B]e {U}e (5) where [B]e is a derivative operator[15] and {U}e is the element nodal displacement vector. Thus, the piezoresistive Eq. (4) becomes {ρ}e = ρ0 {{Im } + [Π] [D] [B]e {U}e }

(6)

Figure 1: Applied loads and boundary conditions for a domain that includes piezoresistive material. Essentially, two FE problems must be considered. The first one is the traditional mechanical FEM [15]: [K] {U} = {F} (7) where [K] is the stiffness matrix, {U} is the nodal displacement vector, and {F} is the nodal load vector. The second FEM problem is the electrical conduction problem whose equilibrium equation is given by [15] [C] {Φ} = {I} (8) where [C] is the conductivity matrix, {Φ} is the nodal electric potential vector, and {I} is nodal electrical current vector. This problem is considered only for the piezoresistive medium. The electrical degrees of freedom are also approximated using FEM shape functions [15] φe ≃ [N]e {Φ}e ⇒ ▽φe = ▽ ([N]e {Φ}e ) = [Bφ ]e {Φ}e

(9)

The matrix [C] is given by[15] [C] =

Ne X

[C]e and [C]e =

Z

Ωe

e=1

t

[Bφ ]e [k]e [Bφ ]e dΩe

(10)

where Ne is the number of piezoresistive finite elements in the domain and [k]e is the conductivity coefficient matrix defined by Eq. (2), that is [k]e = [ρ]−1 e

(11)

Thus, based on the previous equations, a procedure to solve the piezoresistive problem is described[2, 4]: 3

1. Solve traditional FEM Eq. (7) (linear or nonlinear) considering the applied loads to the load cell; 2. Calculate the conductivity matrix using Eq. (2) and Eq. (6) based on the FEM nodal displacements obtained previously; 3. Solve the electrical conduction FEM problem Eq. (8) to obtain the nodal electrical potentials. Now, usually due to performance reasons, to amplify the electric potential change and to compensate temperature effects, the piezoresistive elements (such as strain-gages, for example) operate in a “Wheatstone bridge” configuration as shown in figure 1a.

a)

b)

c)

Figure 2: a) Usual “Wheatstone bridge”; b) and c) Possible “Wheatstone bridge” configurations for a piezoresistive material Considering a continuum piezoresistive element a variation of the previous “Wheatstone bridge” configuration is used, as shown in figures 1b and 1c [2]. The value of (φout = φout2 − φout1 ) are obtained directly from the solution of the FEM equation 8 for the piezoresistive medium. In this work, the material angle is fixed, however, it could also be considered a design variable in the optimization problem. 5. Topology Optimization Formulation 5.1. Concept Topology optimization is based on two main concepts[16]: the extended design domain and the relaxation of the design domain. The extended design domain is a large fixed domain that must contain the whole structure to be determined by the optimization procedure. The objective of topology optimization is to determine the holes and connectivities of the structure by finding the optimal distribution of material in this domain. A main question in topology optimization is how to change the material from zero to one. To avoid an ill-posed problem caused by a discrete material distribution function (0-1), the optimization problem must be relaxed by allowing the material to assume intermediate property values during the optimization, which can be achieved by defining a material model [16]. Essentially, the material model approximates the material distribution by defining a function of a continuous parameter (design variable) that determines a mixture of two materials throughout the domain allowing the appearance of intermediate (or composite) materials - rather than only void or full material - in the final solution. In this work, a topology optimization material model based on the so-called SIMP (“Solid Isotropic Material with Penalization”) will be employed together with a filtering technique to control the mesh dependency and checkerboard problems[16, 17]. The traditional SIMP material model states that in each point of the domain, the material property is given by [16]: D H = γ p D0

(12)

where DH and D0 are the Young modulus of the homogenized material and basic material, respectively, that will be distributed in the domain, γ is a pseudo-density describing the amount of material in each point of the domain which can assume values between 0 and 1, and p is a penalization factor to recover the discrete nature of the design. For γ equal to 0 the material is equal to void and for γ equal to 1 the 4

material is equal to solid material. Since the piezoresistive material is kept fixed in the design domain, this material model is applied only to the material of coupling structure domain. 5.2. Optimization Problem Formulation The piezoresistive force sensor can be understood as a compliant mechanism that converts the input force into local output displacement, and so strain and stress, in the piezoresistive element which converts it into output voltage due to the resistivity change. The objective function of the piezoresistive load cell design problem is to maximize the “Wheatstone bridge” output voltage difference (∆φ = φout2 − φout1 ) for a certain applied force to the load cell. However, following the idea of the compliant mechanism design[18, 19], if only the force sensor response is considered a structure with very low stiffness may be obtained. Thus, some stiffness must be provided to the force sensor structure to support the applied force. This can be achieved by minimizing the mean compliance C given by [16]: t

t

C = {U} {F} = {U} [K] {U}

(13)

Both objective functions are combined by defining the following logarithmic function: F = w log(∆φ) −(1 − w) logC

(14)

where w is a weight coefficient that allows us to control the trade-off between force sensor stiffness and response. In this work, the location of piezoresistive material is kept fixed, and the objective is to find the structural topology that gives the highest output voltage difference. Thus, considering the FEM discretization, the continuous form of the final design optimization problem is defined in the following way: Max F γ R (15) Such that V = Ω γdΩ ≤ V0 Equilibrium Equations where the only constraint is a volume material constraint which controls the amount of material (not including piezoresistive material) in the force sensor design domain. The filtering technique allows us to control the mesh dependency and checkerboard which are well-known problems in topology optimization[16, 17]. By considering the discretized FEM problem, the same optimization problem is defined as: Max F γe PNe (16) γe Ve ≤ V0 Such that V = e=1 [K] {U} = {F} and [C] {Φ} = {I} filtering technique

6. Numerical Implementation Four-node isoparametric bilinear elements under plane stress assumption with two degrees of freedom per node (ux and uy )[15] were applied to model the elastic domain in the mechanical problem including all elements (piezoresistive and non-piezoresistive elements). The electrical problem is solved only for the piezoresistive elements by using four-node isoparametric bilinear elements with one degree of freedom per node (Φ). A flow chart of the optimization algorithm describing the steps involved is shown in figure 3. The software was implemented in C language. The design variables are the pseudo-density γe defined only in the elastic domain (the piezoresistive is out of the design domain) which can assume different values in each finite element. In this work, the mathematical programming method called Sequential Linear Programming (SLP) [20, 21] is applied to solve the optimization problem. Suitable moving limits are introduced to assure that the design variables do not change by more than 5 − 15% between consecutive iterations[21]. A new set

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Figure 3: Flow chart of optimization procedure. of design variables γe is obtained after each iteration, and the optimization continues until convergence is achieved for the objective function. The initial guess for design variables γe consists of uniform values. The linearization of the problem (Taylor series) in each iteration requires the sensitivities (gradients) of the multi-objective function and constraints in relation to the design variables γe . The sensitivity of volume constraint defined in Eq. (16) is straightforward to calculate. The sensitivity of objective function is obtained by deriving Eq. (14) in relation to γe resulting in an expression that is a function of the sensitivity of output voltage (∆φ = φout2 − φout1 ) and mean compliance (C). The sensitivity of mean compliance is well-known in the literature[16]. The sensitivity of output voltage was derived and it is equal to: d [C] d∆φ = − {M}t [C]−1 {Φ} dγe dγe

(17)

where vector {M} is a vector that has all terms equal to zero, but those terms related to the electrical degrees of freedom φout1 and φout2 , which are equal to 1. The sensitivity d[C] dγe can be obtained by deriving Eqs. (10) and (11) which gives Z d[C] d[C]e t d [k]e = = [Bφ ]e [Bφ ]e dΩe (18) dγe dγe dγe Ωe where

d [k]e [K]e t = [k]e ρ0 [Π] [D] [B]e [K]−1 {U} [k]e dγe γe

(19)

To avoid the mesh dependency and checkerboard problem a filter [22] was implemented. Essentially, the filter calculates the upper and lower bounds of the design variable γe based on the design variable values of neighbouring elements, avoiding strong material gradients inside of the domain. The amount of neighbouring elements considered in the calculation is defined by specifying a radius. The filter is mesh independent[22]. Regarding material model defined by Eq. (12), a continuation approach is adopted for p coefficient value[23], that is, p values starts equal to 1 and it is gradually increased along the iterations up to 3. The reason is that the optimization problem with p equal to 3 has many local minima, thus, by using the continuation approach makes the optimization algorithm to stop in a more appropriate local minimum.

7. Results Table 1 presents the piezoresistive material n − Si properties ρ0 , π11 , π12 , π44 , D11 , D12 , and D44 used in the simulations for all examples. The piezoresistive material is orthotropic and the angle between its

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axes and the global axes is equal to θ. The elastic medium is made of Aluminum which is isotropic with Young’s modulus and Poisson’s ratio equal to 80 GPa and 0.33, respectively. Table 1: Piezoresistive material properties (n − Si) ρ0 Ω.m 0.117

π11 10−11 m2 /N -102.2

π12 10−11 m2 /N 53.4

π44 10−11 m2 /N -13.6

D11 GP a 165.7

D12 GP a 63.9

D44 GP a 79.6

The voltage applied to the “Wheatstone bridge” for all examples is equal to 50V . When the optimization process is complete, the result is a material distribution over the mesh with some intermediate values of density (“gray scale”) that represents the presence of some intermediate material. The final interpretation is achieved by simply applying a threshold value to average density values.

a)

b)

Figure 4: Initial design domains considered for piezoresistive force and torque sensors design. All initial design domains consist of a domain of piezoresistive material that remain unchanged during the optimization and a domain S of Aluminum where the optimization is conducted. A layer of Aluminum material surrounding the piezoresistive material is also kept fixed to guarantee that the piezoresistive material will be fully attached to the structure. The total volume constraint of the material V0 is considered to be 50% of the volume of the whole domain without piezoresistive material (domain S) for both examples. The weight coefficient w and penalization coefficient p are specified to be equal to 0.7 and 3, respectively. Both domains were discretized into 1.250 four-node isoparametric bilinear finite elements (rectangle discretized by a 25×50 mesh). A filter with radius equal to 2 mm was applied[22]. In the first example, the design of a cantilever-type force sensor is considered. The initial design domain considered is described in figure 4a. The mechanical and electrical boundary conditions are shown in the same figure. The θ angle is equal to 30◦ . The optimization problem is defined as the maximization of the output voltage when a force equal to 500 N is applied to upper right corner, as shown in figure 4a. The topology optimization in figures 5a and 5b with θ equal to 0◦ and 30◦ respectively. The output voltage obtained is equal to 6.4mV . Considering the entire domain full of Aluminum the output voltage would be 0.04mV , thus there was a 25 times improvement in the force sensor response. In the second example, the design of a torque sensor based on the initial design domain described in figure 4b is considered. The mechanical and electrical boundary conditions are shown in the same figure. The optimization problem is defined as the maximization of the output voltage when a torque equal to 2.10-3 N.m is applied to the right edge, as shown in figure 4b. The topology optimization result is shown in figure 6 with corresponding objective function convergence curve. The output voltage obtained is equal to 11.2mV . Considering the entire domain full of Aluminum the output voltage would be 6.08mV , thus there was a 2 times improvement in the torque sensor response. 7

a)

b)

Figure 5: a) Topology optimization result θ = 0◦ ; b) Corresponding result θ = 30◦ .

a)

b)

Figure 6: a) Topology optimization result; b) Corresponding objective function convergence curve.

8. Conclusions The design of piezoresistive sensors using topology optimization was presented. The optimization problem was defined as a the design of an optimum flexible structure that bonded to the piezoresistive material generates the maximum response in terms of resistivity change (or output voltage). To illustrate the problem some design of piezoresistive force and torque sensors were shown. The obtained sensor designs presented higher response than simple designs, showing that topology optimization can be a helpful tool to design these devices. As a future work, designs of other types of piezoresistive sensors can be considered such as accelerometers, inclinometers, and membrane pressure sensors. 9. Acknowledgments The first author thanks FAPESP (S˜ao Paulo State Foundation Research Agency) for supporting him through project no 02/04533 − 9. First and last authors acknowledge “Japan Society for the Promotion of Science” (JSPS) for supporting this research.

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Topology Optimization - Theory, Methods and Applications.

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