Computer-implemented method for optimizing geometric nonlinearities of a sensor geometry

Abstract

Computer-implemented method (100) for optimizing geometric nonlinearities of a sensor geometry (10), wherein an initial FEM model (11) of a sensor geometry (10) comprising a plurality of finite elements e (12) is created, wherein the finite elements e are formed from nodes (14) connected by line segments (13), wherein each node (14) is assigned a set of coordinates xi,0, wherein design parameters r of the sensor geometry (10) and displacements Δxi(r) (15) of the coordinates xi,0 of at least a subset of the nodes (14) are defined as functions of the design parameters r, wherein in an optimization loop k eigenfrequencies f and eigenvectors ϕ of the sensor geometry (10) are determined, wherein values for cubic nonlinear coupling coefficients αn,m,l and for total derivatives dαn,m,ldr for the Natural frequencies f and / or eigenvectors ϕ are determined, where values of the design parameters r, which optimize a target function,are determined, and where new coordinates xi(r) = xi,0+ Δxi(r) are determined at least for the subset of nodes (14) by applying the displacements Δxi(r) evaluated for the values of the design parameters r to the coordinates xi,0 of the nodes.,

Description

State of the art

[0001] The present invention relates to a computer-implemented method for optimizing geometric nonlinearities of a sensor geometry and a manufacturing method for a sensor.

[0002] MEMS resonators, among others, are used in various miniaturized sensors and actuators. Such MEMS resonators are characterized by the geometric design of the mechanical oscillator, which is tailored to the specific application. In MEMS angular rate sensors in particular, the design of the MEMS resonator is very complex and delicate. Here, the sensor geometries undergo a complex development process until a satisfactory result is achieved.

[0003] Commercially available MEMS angular rate sensors, as well as many other MEMS resonators, are essentially composed of rectangular structures whose lengths and widths are selected to ensure optimal resonator performance with respect to application-specific parameters. The initial design is created by a human designer and parameterized based on the lengths and widths of the structural elements. Finite element models (FEM models) are used for subsequent design phases, based on which the sensor geometry is evaluated and application-specific parameters that describe the product quality are assessed.

[0004] In many applications, MEMS resonators oscillate with amplitudes that are large compared to the resonator's geometric dimensions. Geometric nonlinearities can occur, significantly influencing the resonator's dynamic behavior. The resulting effects can, for example, lead to instabilities in angular rate sensors or even the breakage of micromirrors. Such effects result in nonlinear couplings between different mechanical vibration modes of the resonator, which can lead to the aforementioned undesirable effects. The nonlinear coupling coefficients, which couple the equations of motion of the vibration modes and result from the geometric nonlinearities, arise purely from the resonator's geometry and can be calculated using FEM methods. Disclosure of the invention

[0005] The present invention is based on the object of providing a method with which geometric nonlinearities of a sensor geometry can be optimized.

[0006] To achieve the object underlying the invention, a computer-implemented method for optimizing geometric nonlinearities of a sensor geometry is proposed, comprising the steps - Creating an initial FEM model of a sensor geometry, wherein the initial FEM model comprises a plurality of finite elements e formed from nodes connected by line segments, each node being assigned a set of coordinates x i,0 is assigned, - Defining design parameters r of the sensor geometry, - Defining displacements Δx i (r) of the coordinates x i,0 at least a subset of the nodes as functions of the design parameters, - Determine the partial derivatives ∂xl→∂r the displacements Δx i (r) according to the design parameters, - Representation of cubic nonlinear coupling coefficients α n,m,l = α n,m,l (r, Φ) as a function of at least the design parameters r and eigenvectors ϕ to eigenmodes, and preferably eigenfrequencies f, of the sensor geometry, where the cubic nonlinear coupling coefficients α n,m,l Coupling the equations of motion of the eigenmodes n,m,l of the sensor geometry, - Perform an optimization loop with the steps: ◯ Determination of k eigenfrequencies f and eigenvectors ϕ of the sensor geometry, where k ≥ max {n,m,l}, ◯ Determination of values for the cubic nonlinear coupling coefficients α n,m,l for the eigenfrequencies f and / or eigenvectors ϕ, ◯ Determination of total derivatives dαn,m,ldr the cubic nonlinear coupling coefficients according to the design parameters r, ◯ Determination of values of the design parameters r using an optimization algorithm, which optimizes an objective function, where the values for the cubic nonlinear coupling coefficients α n,m,l and the total derivatives dan,m,ldr the cubic nonlinear coupling coefficients of the objective function and / or are taken into account as constraints when implementing the optimization algorithm, ◯ Determine new coordinates x i (r) = x i,0 + Δx i (r) at least for the subset of nodes by applying the displacements Δx evaluated for the values of the design parameters r i (r) to the coordinates x i,0 the nodes, - Repeat the optimization loop until a termination criterion is met.

[0007] The FEM model consists of a multitude of finite elements e. The finite elements e are formed by the nodes and the line segments connecting them. Each of the nodes i is assigned a set of coordinates x i,0 assigned, where x i,0 represents the entire set of coordinates of the respective node i. If the FEM model is a two-dimensional FEM model, then x i,0 representing the set of x and y coordinates (x, y) i,0 .

[0008] The initial FEM model is preferably created using known methods. The design parameters r can, in particular, be geometric design parameters r of the sensor geometry to be created. In addition, displacements Δx i (r) of the coordinates x i,0 at least a subset of the nodes as a function of the design parameters r. By applying the displacements Δxi (r) to the coordinate x i,0 of the i-th node according to x i (r) = x i,0 + Δx i (r) new coordinates x i In other words, the coordinates of at least a subset of the nodes are defined as a function of the design parameters. A design parameter r can be defined in such a way that it influences individual or multiple nodes.

[0009] For each of the nodes, the partial derivatives ∂xl→∂r determined according to the design parameters r. The derivatives can be determined analytically or numerically.

[0010] The method also includes the cubic nonlinear coupling coefficients α n,m,l as a function of at least the design parameters r and the eigenvectors ϕ to eigenmodes, and preferably the eigenfrequencies f, of the sensor geometry.

[0011] According to the method, an optimization loop is then run through, whereby in a first step of the optimization loop, k eigenfrequencies f and eigenvectors ϕ of the sensor geometry are determined. Here, k ≥ max {n,m,l}, i.e., k is chosen so large that the modes whose nonlinear coupling coefficients α n,m,l to be optimized. Preferably, k is of the order of 10 1 up to 10 2 . For the determined eigenfrequencies f and eigenvectors ϕ, the cubic nonlinear coupling coefficients are evaluated. Furthermore, the total derivatives dαn,m,ldr the cubic nonlinear coupling coefficients are determined according to the design parameters r.

[0012] In a next step of the optimization loop, the cubic nonlinear coupling coefficients α evaluated for the determined eigenfrequencies f and eigenvectors ϕ are calculated. n,m,las well as the total derivatives dαn,m,ldr the cubic nonlinear coupling coefficients for all design parameters r are passed to an optimization algorithm to optimize an objective function. The values for the cubic nonlinear coupling coefficients α n,m,l and the total derivatives dαn,m,ldr the cubic nonlinear coupling coefficients of the objective function or are taken into account as constraints for the optimization algorithm.

[0013] For example, a constraint can be specified that the magnitude of one, several, or all cubic nonlinear coupling coefficients should be smaller than a given value. By optimizing the objective function, new values for each of the design parameters r can then be determined. In a subsequent step, new coordinates x i (r) = x i,0 + Δx i(r) at least for the subset of nodes by applying the displacements Δx evaluated for the values of the design parameters r i (r) to the coordinates x i,0 of the nodes and thus shift the nodes of the subset of nodes. The optimization loop is then repeated until a termination criterion is met. The termination criterion can also be applied immediately after the step of determining values for the cubic nonlinear coupling coefficients α n,m,l for the eigenfrequencies f and / or eigenvectors ϕ.

[0014] The displacements Δx i (r) of the coordinates can depend linearly on the design parameters. However, other functional dependencies of the displacements Δx i (r) of the design parameters r possible.

[0015] Furthermore, it can be provided that a certain design parameter r determines the coordinates x i,0of all nodes of the subset of nodes, or it can be provided that a design parameter r only affects the coordinates x i,0 one or fewer nodes of the subset of nodes.

[0016] Preferably, the optimization algorithm is a gradient-based optimization algorithm, particularly preferably a sequential quadratic programming (SQP) algorithm or a method of moving asymptotes (MMA) algorithm.

[0017] With further advantage, it can be provided that the total derivatives dαn,m,ldr the cubic nonlinear coupling coefficients according to the design parameters r can be determined by an adjoint sensitivity analysis.

[0018] With even further advantage, it can be provided that the subset of nodes comprises edge nodes of the FEM model or consists of edge nodes of the FEM model.

[0019] This means in particular that only the boundary nodes that discretize the edge of the sensor geometry are made dependent on the design parameters r, or that only for the boundary nodes displacements Δx i (r) of the coordinates as a function of the design parameters.

[0020] Since the inner nodes of the sensor geometry in particular are not dependent on the design parameters r, the computing time can be significantly reduced.

[0021] Furthermore, it can be provided that the optimization loop comprises the step of determining new coordinates for at least some of the nodes of the FEM model that do not belong to the subset of the nodes, wherein the new coordinates are preferably determined by means of an elastic model of the sensor geometry, wherein more preferably the new coordinates of the nodes of the subset of the nodes are taken into account as boundary conditions, in particular as Dirichlet boundary conditions.

[0022] After the new coordinates x i (r) = x i,0 + Δx i(r) for the subset of nodes, in particular for the edge nodes, of the FEM model, the positions of the nodes not belonging to the subset, in particular the inner nodes, of the FEM model are adjusted. An elastic model of the sensor geometry can be used to track the nodes not belonging to the subset, in particular the inner nodes. The tracking of the inner nodes can, in particular, be simply proportional to the displacements Δx i (r) of the nodes belonging to the subset are determined.

[0023] Advantageously, it can be provided that the cubic nonlinear coupling coefficients are given by αn,m,l=αn,m,l(r,ϕ→n,ϕ→m,ϕ→l)=∑e∑p[(ϕ→ne)T(B1e(r,p))TDeB2e(r,p,ϕ→m)ϕ→le+(ϕ→me)T( B1e(r,p))TDeB2e(r,p,ϕ→n)ϕ→le+(ϕ→le)T(B1e(r,p))TDeB2e(r,p,ϕ→n)ϕ→me)⋅det(Je(r,p))] where r is a design parameter, where ϕ→n,ϕ→m,ϕ→l Eigenvectors of the eigenmodes n,m,l are, where e is an index of the finite elements, where p is an integration point index, where D e is a matrix containing material parameters for the finite element e, where B1e(r,p) and B2e(r,p,ϕ→i) and Je(r,p) Matrices for the finite element e are those resulting from the FEM model of the sensor geometry.

[0024] Furthermore, it can be provided that the cubic nonlinear coupling coefficients are normalized according to the following rule: αn,m,l→{αn,m,l;if n,m,l are three distinct indicesαn,m,l / 2;if an index appears twice in n,m,lαn,m,l / 6; if n=m=l

[0025] In principle, any cubic nonlinear coupling coefficient α n,m,ldepend on all design parameters r. Formula (1) specifies the explicit dependencies on the design parameters r. Implicitly, the eigenvectors ϕ also depend on the design parameters r.

[0026] In formula (1), the sum is calculated over all finite elements e. The index on the matrices and vectors indicates that these must be evaluated individually for each finite element e. For each finite element e, the sum is calculated over the integration points, which are indexed by the integration point index p. The material parameters of the matrices D e depend on the material, e.g. silicon, from which the finite element e is made. The matrices B1e(r,p) and B2e(r,p,ϕ→i) and J e (r, p) are matrices that are specific to each finite element e and also depend on the integration point p and the design parameters r. The matrices B2e(r,p,ϕ→i) also depend on the eigenvectors ϕi the eigenmodes.

[0027] The formula (1) can be obtained in the following way:

[0028] The time-dependent shift u→ (time-dependent deflection of the mechanical structure) under the influence of external forces can be approximated as a superposition of the first n mechanical eigenmodes: u→(t)≈∑i=1kqi(t)ϕ→i

[0029] Here t is the time and q i the modal amplitude of the eigenmode i. In the case of geometric nonlinearities, the so-called strain energy (potential energy) U of the mechanical structure can be written as: U=∫VS(u→):E(u→)dV

[0030] The integrand is the element-wise scalar product of the matrices S and E. S is the second Piola-Kirchhoff stress tensor, E is the Green-Lagrange strain tensor, and V is the volume of the sensor geometry. Furthermore, linear elastic material behavior is assumed. S and E then contain both linear and quadratic terms in u→. Inserting equation (3) into (4) and collecting the terms that are cubic in the eigenvectors ϕ allows the cubic nonlinear coupling coefficient α n,m,l , which couples the modes n, m and / , according to (1).

[0031] It is preferably provided that the optimization loop comprises the step of determining a mass matrix M and / or a stiffness matrix K of the sensor geometry, and that the natural frequencies f and eigenvectors ϕ of the sensor geometry are determined from the mass matrix M and / or the stiffness matrix K, in particular by solving an eigenvalue problem for the mass matrix M and / or the stiffness matrix K.

[0032] The eigenfrequencies f and eigenvectors ϕ of the sensor geometry can be determined by solving the eigenvalue problem (K−ωi2M)ϕ→i=0→ determine, where ω i = 2πf i with the natural frequency f i is the eigenmode i and where ϕ→i is the eigenvector of the eigenmode i.

[0033] The mass matrix M and the stiffness matrix K are the linear FEM system matrices, and their dimensions correspond to the mechanical degrees of freedom of the discretized sensor geometry. When determining the eigenvectors, it must be considered that they are mass-normalized: ϕ→iTMϕ→i=1.

[0034] Preferably, the total derivatives of the cubic nonlinear coupling coefficients with respect to the design parameters r are given by dαn,m,ldr=∂αn,m,l∂r+∑j∑e[(λ→je)T(∂Ke∂r−ωj2∂Me∂r)ϕ→je+12ηj(ϕ→je)T∂Me∂rϕ→je] where λ→j and η j are adjoint variables.

[0035] The summation index j runs over the unique eigenmodes n, m, l. If, for example, n = m then j = n and j = l are evaluated and summed. The derivatives of the system matrices K and M are calculated for each element e based on ∂x→∂r calculated using ∂Ke∂r=∑p[(∂B1e(r,p)∂r)TDeB1e(r,p)⋅det(Je(r,p))+(B1e(r,p))TDe∂ B1e(r,p)∂r⋅det(Je(r,p))+(B1e(r,p))TDeB1e(r,p)⋅∂det(je(r,p))∂r] ∂Me∂r=∑p[ρeNTN⋅∂det(je(r,p))∂r] where ρ e is the density of element e and N is a matrix resulting from the FEM discretization of the sensor geometry.

[0036] The derivation ∂∂r takes into account the derivative according to the explicit dependence on the design parameters r (in contrast, GDR as a total derivative also the implicit dependence on the eigenvectors ϕ, which also depend on the design parameters r) and is based on ∂x→∂r calculated. ∂αn,m,l∂r is calculated according to: ∂αn,m,l∂r=∑e∑p[((ϕ→ne)T(∂B1e(r,p)∂r)TDeB2e(r,p,ϕ →m)ϕ→le+(ϕ→ne)T(B1e(r,p))TDe∂B2e(r,p,ϕ→m)∂rϕ→le+ (ϕ→me)T(∂B1e(r,p)∂r)TDeB2e(r,p,ϕ→n)ϕ→le+(ϕ→me)T( B1e(r,p))TDe∂B2e(r,p,ϕ→n)∂rϕ→le+(ϕ→le)T(∂B1e(r,p )∂r)TDeB2e(r,p,ϕ→n)ϕ→me+(ϕ→le)T(B1e(r,p))TDe∂B2e(r,p,ϕ→n)∂rϕ→me)⋅det(Je(r,p))+((ϕ→ne)T(B1e(r,p)) TDeB2e(r,p,ϕ→m)ϕ→le+(ϕ→me)T(B1e(r,p))TDeB2e(r,p, ϕ→n)ϕ→le+(ϕ→le)T(B1e(r,p))TDeB2e(r,p,ϕ→n)ϕ→me)⋅∂ det(Je(r,p))∂r]

[0037] Afterwards ∂αn,m,i∂r just like α n,m,l normalized in equation (2). λ→j and η j are the adjoint variables. They must be calculated for each eigenmode j, of which α n,m,l depends, can be calculated once (independent of the number of design parameters). η j is calculated according to: ηj=−ϕ→jT∂αn,m,l∂ϕ→j

[0038] Depending on whether j = n, j = m or j = l: ∂αn,m,l∂ϕ→n=Σe(Le)T∑p[((B1e(r,p))TDeB2e(r,p,ϕ→m)ϕ→le+(B2e(r,p, ϕ→l))TDeB1e(r,p)ϕ→me+(B2e(r,p,ϕ→))TDeB1e(r,p)ϕ→le)⋅det(Je(r,p)) ∂αn,m,l∂ϕ→n=Σe(Le)T∑p[((B1e(r,p))TDeB2e(r,p,ϕ→l)ϕ→ne+(B2e(r,p,ϕ →n))TDeB1e(r,p)ϕ→le+(B2e(r,p,ϕ→l))TDeB1e(r,p)ϕ→ne)⋅det(Je(r,p))) ∂αn,m,l∂ϕ→l=Σe(Le)T∑p[((B1e(r,p))TDeB2e(r,p,ϕ→m)ϕ→me+(B2e(r,p,ϕ→ n))TDeB1e(r,p)ϕ→me)⋅(B2e(r,p,ϕ→m))TDeB1e(r,p)ϕ→ne)⋅det(Je(r,p))]

[0039] Here, L e a matrix resulting from the connectivity of the FEM mesh and assigning the degrees of freedom of element e to the corresponding global degrees of freedom.

[0040] An additional normalization is then carried out in the following cases: If n = m = l then ∂αn,m,l∂ϕj→→12∂αn,m,l∂ϕ→j. If exactly two indices of n, m, l are identical, only for the third (the single) index ∂αn,m,l∂ϕ→j→12∂αn,m,l∂ϕ→j.

[0041] A global λ→j is then determined from the solution of the following system of equations: (K−ωj2M)λ→j=−∂αn,m,l∂ϕj→−ηjMϕ→jϕ→jTMλ→j=0

[0042] The adjoint variable for each element can then be determined using λ→je=Leλ→j.

[0043] Thus, equation (5) can be evaluated and the calculation of the total derivative dαn,m,ldr be completed.

[0044] With further advantage, it can be provided that the method further comprises the step of defining geometric boundary conditions for the sensor geometry, wherein preferably the objective function is further a function of the boundary conditions for the sensor geometry.

[0045] By taking geometric boundary conditions into account, it can be ensured that the sensor geometry meets specified technical requirements.

[0046] In particular, the geometric boundary conditions can include minimum widths t(r) and / or minimum distances d(r) between elements or regions of the sensor geometry. The boundary conditions, in particular the minimum widths t(r) and / or the minimum distances d(r), depend on the coordinates of the nodes and thus, at least indirectly, also on the design parameters r.

[0047] For stability reasons, it may be stipulated that certain elements of the sensor geometry, such as beams or springs, must not be less than a certain minimum width t(r) or must have a certain minimum distance d(r) from other elements, such as beams or springs.

[0048] During the optimization loop, it is possible to check whether the geometric constraints, in particular the minimum widths t(r) and minimum distances d(r), are met. If the geometric constraints are not met, the values of the design parameters r determined by the optimization algorithm can be discarded.

[0049] To determine whether the boundary conditions are met, the relevant parameters, such as the widths and spacing of elements or regions of the sensor geometry, can be determined. In particular, it can be provided that the widths or spacing of the elements are determined using a ray tracing method.

[0050] In a ray tracing technique, the intersection point of an inward or outward normal vector with the nearest line segment is determined. To speed up the process of finding the nearest line segment intersected by the normal vector, a kd-tree can be used to significantly reduce the number of line segments to be searched.

[0051] It is preferably provided that the design parameters r comprise lengths and / or widths and / or rounding radii of elements or regions of the sensor geometry, and / or sizes and / or shapes of perforation holes in elements and / or regions of the sensor geometry, and / or a global layer thickness of the sensor geometry and / or local variations of a layer thickness of the sensor geometry, and / or bending of the sensor geometry, and / or local variations of etching losses of a manufacturing process for a sensor.

[0052] If the FEM model is a two-dimensional FEM model, the global layer thickness can, in particular, be a layer thickness perpendicular to the two-dimensional plane of the sensor geometry. Sensor deflections can be caused by intrinsic stress in the material and can be accounted for via a corresponding design parameter in the sensor geometry optimization process. Etching losses are to be expected during the manufacturing process for the sensor, which will later be manufactured based on the calculated sensor geometry. Such etching losses can also be considered as design parameters for the optimization of the sensor geometry.

[0053] It is further advantageous that the sensor geometry is a sensor geometry of a MEMS sensor, in particular a MEMS yaw rate sensor.

[0054] With even further advantage, it can be provided that the sensor geometry is a 2D geometry.

[0055] In principle, the process can be used to optimize geometric nonlinearities of three-dimensional sensor geometries. Modern MEMS sensors, especially modern MEMS angular rate sensors, are often manufactured using an etching process along a single axis. Accordingly, the optimization of geometric nonlinearities can also be performed based on a two-dimensional FEM model, which is then extruded for sensor production.

[0056] A further solution to the problem underlying the invention consists in a method for producing a sensor, in particular a MEMS sensor, further in particular a MEMS yaw rate sensor, wherein a sensor geometry was created by means of a previously described computer-implemented method for optimizing geometric nonlinearities of sensor geometries, and wherein the sensor is produced on the basis of the optimized sensor geometry, further preferably in an etching process.

[0057] The invention is explained in more detail below with reference to the attached figure.

[0058] The single figure shows a flow chart for a computer-implemented method for optimizing geometric nonlinearities of a sensor geometry.

[0059] The figure illustrates a computer-implemented method 100 for optimizing geometric nonlinearities of a sensor geometry 10 in accordance with the invention. In a first method step S1, an initial FEM model 11 of a sensor geometry 10 is created. The initial FEM model 11 comprises a plurality of finite elements 12, which are formed from nodes 14 connected by line segments 13. Each node 14 is assigned a set of coordinates x i,0 In a second process step S2, design parameters r of the sensor geometry 10 are defined. In addition, displacements Δx i (r) 15 of the coordinates xi,0 of boundary nodes 16 as a function of the design parameters r. A design parameter r can be defined in such a way that it shifts individual or multiple boundary nodes 16. In addition, for each design parameter r, the partial derivatives ∂xι→∂r the displacements Δx i (r) 15. In addition, cubic nonlinear coupling coefficients α n,m,l = α n,m,l (r,ϕ,f) as a function of the design parameters r, of eigenfrequencies f and of eigenvectors ϕ to eigenmodes of the sensor geometry 10, where the cubic nonlinear coupling coefficients α n,m,l Equations of motion of the eigenmodes n,m,l of the sensor geometry 10 are coupled together.

[0060] In a third process step S3, k eigenfrequencies f and eigenvectors ϕ for eigenmodes of the sensor geometry 10 are determined. Subsequently, in a further process step S4, the cubic nonlinear coupling coefficients α n,m,l = α n,m,l (r, ϕ, f) for the design parameters r, the eigenfrequencies f and the eigenvectors ϕ from step S3 are evaluated. In step S4, the total derivatives dαn,m,ldr the cubic nonlinear coupling coefficients are determined according to the design parameters r. In step S5, it is checked whether a termination criterion is met. If the termination criterion is not met, a further method step S6 provides an objective function that defines an optimization goal for the sensor geometry 10. Using a gradient-based optimization algorithm, values of the design parameters r are determined that optimize the objective function. In a further method step S6, new coordinates x i (r) = x i,0 + Δx i(r) is determined for the edge nodes 16. Subsequently, in a step S8, the inner nodes 17 of the sensor geometry 10 are adjusted proportionally to the displacements of the edge nodes 16. The method 100 then continues with method step S3. If the termination criterion is met during the evaluation in method step S5, the method 100 is terminated in a final method step S9.

Claims

[1] Computer-implemented method (100) for optimizing geometric nonlinearities of a sensor geometry (10), comprising the steps - Creating an initial FEM model (11) of a sensor geometry (10), wherein the initial FEM model (11) comprises a plurality of finite elements e (12) which are formed from nodes (14) connected to line segments (13), wherein each node (14) is assigned a set of coordinates x i,0 is assigned, - Defining design parameters r of the sensor geometry (10), - Defining displacements Δx i (r) (15) of the coordinates x i,0 at least a subset of the nodes (14) as functions of the design parameters r, - Determine the partial derivatives ∂xι→∂r the displacements Δx i (r) (15) according to the design parameters, - Representation of cubic nonlinear coupling coefficients α n,m,l = αn,m,l (r, ϕ) as a function of at least the design parameters r and eigenvectors ϕ to eigenmodes, and preferably eigenfrequencies f, of the sensor geometry (10), wherein the cubic nonlinear coupling coefficients α n,m,l Couple the equations of motion of the eigenmodes n,m,l of the sensor geometry (10), - Perform an optimization loop with the steps: ◯ Determination of k eigenfrequencies f and eigenvectors ϕ of the sensor geometry (10), where k ≥ max {n,m,l}, ◯ Determination of values for the cubic nonlinear coupling coefficients α n,m,l for the eigenfrequencies f and / or eigenvectors ϕ, ◯ Determination of total derivatives dαn,m,ldr the cubic nonlinear coupling coefficients according to the design parameters r, ◯ Determination of values of the design parameters r using an optimization algorithm, which optimizes an objective function, where the values for the cubic nonlinear coupling coefficients α n,m,l and the total derivatives dαn,m,ldr the cubic nonlinear coupling coefficients of the objective function and / or are taken into account as constraints when implementing the optimization algorithm, ◯ Determine new coordinates x i (r) = x i,0 + Δx i (r) at least for the subset of nodes (14) by applying the displacements Δx evaluated for the values of the design parameters r i (r) to the coordinates x i,0 the nodes, - Repeat the optimization loop until a termination criterion is met. [2] Computer-implemented method (100) according to claim 1, wherein the optimization algorithm is a gradient-based optimization algorithm, in particular a sequential quadratic programming (SQP) algorithm or a method of moving asymptotes (MMA) algorithm. [3] Computer-implemented method (100) according to claim 1 or 2, wherein the total derivatives dαn,m,ldr the cubic nonlinear coupling coefficients according to the design parameters r can be determined by an adjoint sensitivity analysis. [4] Computer-implemented method (100) according to one of the preceding claims, wherein the subset of the node points (14) comprises edge node points (16) of the FEM model (11) or consists of edge node points (16) of the FEM model (11). [5] Computer-implemented method (100) according to one of the preceding claims, wherein the optimization loop comprises the step of determining new coordinates for at least some of the nodes (17) of the FEM model (11) that do not belong to the subset of the nodes (14), wherein the new coordinates are preferably determined by means of an elastic model of the sensor geometry (10), wherein more preferably the new coordinates of the nodes (14) of the subset of the nodes (14) are taken into account as boundary conditions, in particular as Dirichlet boundary conditions. [6] Computer-implemented method (100) according to one of the preceding claims, wherein the cubic nonlinear coupling coefficients are given by αn,m,l=αn,m,l(r,ϕ→n,ϕ→m,ϕ→l)=∑e∑p[((ϕ→ne)T(B1e(r,p))TDeB2e(r,p,ϕ→m)ϕ→le+(ϕ→me)T( B1e(r,p))TDeB2e(r,p,ϕ→n)ϕ→le+(ϕ→le)T(B1e(r,p))TDeB2e(r,p,ϕ→n)ϕ→me)⋅det(Je(r,p))] where r is a design parameter, where ϕ→n,ϕ→m,ϕ→l Eigenvectors of the eigenmodes n,m,l, where e is an index of the finite elements(12), where p is an integration point index, where D e is a matrix containing material parameters for the finite element e, where B1e(r,p) and B2e(r,p,ϕi→) and J e (r,p) are matrices for the finite element e resulting from the FEM model (11) of the sensor geometry (10). [7] Computer-implemented method (100) according to one of the preceding claims, wherein the optimization loop comprises the step of determining a mass matrix M and / or a stiffness matrix K of the sensor geometry (10), and that the natural frequencies f and eigenvectors ϕ of the sensor geometry (10) are determined from the mass matrix M and / or the stiffness matrix K, in particular by solving an eigenvalue problem for the mass matrix M and / or the stiffness matrix K. [8] Computer-implemented method (100) according to claim 7, wherein the total derivatives of the cubic nonlinear coupling coefficients with respect to the design parameters r are given by dαn,m,ldr=∂αn,m,l∂r+∑j∑e[(λ→je)T(∂Ke∂r−ωj2∂Me∂r)ϕ→je+12ηj(ϕ→je)T∂Me∂rϕ→je] are given, where λj→ and η j are adjoint variables. [9] Computer-implemented method (100) according to one of the preceding claims, wherein the termination criterion is reaching predetermined values of the cubic non-linear coupling coefficients α n,m,l and / or reaching a predetermined threshold value of the objective function and / or reaching a predetermined threshold value of a gradient of the objective function. [10] Method for producing a sensor, in particular a MEMS sensor, further in particular a MEMS yaw rate sensor, wherein a sensor geometry (10) was created by means of a computer-implemented method (100) for optimizing geometric non-linearities of sensor geometries (10) according to one of the preceding claims, and wherein the sensor is produced on the basis of the optimized sensor geometry (10), preferably in an etching process.

Images