Computer-implemented method for optimizing geometric non-linearities of a sensor geometry

Abstract

The invention relates to a computer-implemented method (100) for optimizing geometric non-linearities 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 generated; the finite elements e are formed from node points (14) connected by line segments (13); each node point (14) is assigned a set of coordinates x i , 0 ; design parameters r of the sensor geometry (10) are defined, and displacements Δx i (r) (15) of the coordinates x i , 0 <sb / > of at least one subset of node points (14) are defined, as functions of the design parameters r; natural frequencies f and eigenvectors ϕ of the sensor geometry (10) are determined in an optimization loop k; values for cubic non-linear coupling coefficients α n , m, l <sb / > and for total derivatives d α n,m,l / dr are determined for the natural frequencies f and / or eigenvectors ϕ; design parameter r values which optimize a target function are determined; and new coordinates x i(r) = x i ,0 + Δx i (r) are determined at least for the subset of node points (14) by applying the displacements Δx i (r) defined for the values of the design parameters r to the coordinates x i , 0 of the node points.

Description

[0001] R. 410718 - 1 -Description Title Computer-implemented method for optimizing geometric nonlinearities of a sensor geometry State of the art The present invention relates to a computer-implemented method for optimizing geometric nonlinearities of a sensor geometry and to a manufacturing method for a sensor. 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 respective application. Especially in MEMS angular rate sensors, 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 obtained.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. R. 410718 -. 2 -In many applications, MEMS resonators oscillate with amplitudes that are large compared to the resonator's geometric dimensions. This can lead to so-called geometric nonlinearities that significantly influence 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 The object of the present invention is to provide a method with which geometric nonlinearities of a sensor geometry can be optimized. To achieve the object underlying the invention, a computer-implemented method for optimizing geometric nonlinearities of a sensor geometry is proposed, comprising the steps of - creating an initial FEM model of a sensor geometry, wherein the initial FEM model comprises a plurality of finite elements ^ which are formed from nodes connected by line segments, wherein each node is assigned a set of coordinates ^. ^,^ is assigned,- defining design parameters ^ of the sensor geometry,- defining displacements Δ^^(^) of the coordinates ^^,^ of at least a subset of the node points as functions of the design parameters,- determining the partial derivatives^^^^ ^⃗ ^^of the displacements Δ^^(^) according to the design parameters,- Representation of cubic nonlinear coupling coefficients ^^,^,^ = as a function of at least the design parameters ^ and eigenvectors ^ to eigenmodes, and preferably eigenfrequencies ^, of the sensor geometry, where the R. 410718 - 3 - cubic nonlinear coupling coefficients ^ ^,^,^ Couple the equations of motion of the eigenmodes n, m, l of the sensor geometry, - Perform an optimization loop with the steps: o Determine k eigenfrequencies ^ and eigenvectors ^ of the sensor geometry, where k ≥ max {n, m, l}, o Determine values ​​for the cubic nonlinear coupling coefficients ^^,^,^ for the eigenfrequencies ^ and / or eigenvectors ^, o Determine total derivatives ^^ ^,^,^^^ of the cubic nonlinear coupling coefficients according to the design parameters ^, oDetermining values ​​of the design parameters ^ using an optimization algorithm, which optimizes an objective function, whereby the values ​​for the cubic nonlinear coupling coefficients ^ ^,^,^ and the total derivatives ^^ ^,^,^ ^^ of the cubic nonlinear coupling coefficients of the objective function are passed and / or taken into account as constraints when implementing the optimization algorithm, oDetermining new coordinates ^^(^) = ^^,^ + Δ^^(^) at least for the subset of the nodes by applying the displacements Δ^^(^) evaluated for the values ​​of the design parameters ^ to the coordinates ^ ^,^of the nodes,- Repeating the optimization loop until a termination criterion is met. The FEM model consists of a large number of finite elements ^. The finite elements^ are formed by the nodes and the line segments connecting them. Each of the nodes i is assigned a set of coordinates ^ ^,^ assigned, where ^ ^,^ represents the entire set of coordinates of the respective node i. If the FEM model is a two-dimensional FEM model, then ^ ^,^representing the set of x and y coordinates (^, ^)^,^ . The initial FEM model is preferably created using known methods. The design parameters ^ can, in particular, be geometric design parameters^ of the sensor geometry to be created. In addition, displacements Δ^^(^) of the coordinates ^^,^ of at least a subset of the nodes are defined as a function of the design parameters ^. By applying the displacements Δ^^(^) to the coordinate ^^,^ of the i-th node according to ^^(^) = ^^,^ + Δ^^(^), R. 410718 - 4 - the i-th node new coordinates ^ ^ 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 ^ can be defined in such a way that it influences individual or multiple nodes. ^^^^ ^⃗ For each of the nodes, the partial derivatives ^^ with respect to the design parameters ^ are determined. The derivatives can be determined analytically or numerically. Within the framework of the method, the cubic nonlinear coupling coefficients ^^,^,^ are also represented as a function of at least the design parameters ^ and the eigenvectors ^ to eigenmodes, and preferably the eigenfrequencies ^, of the sensor geometry. According to the method, an optimization loop is then run through, whereby in a first step of the optimization loop, k eigenfrequencies ^ and eigenvectors ^ 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 ^ ^,^,^are to be optimized. Preferably, k is of the order of 101 to 102. For the determined eigenfrequencies ^ and eigenvectors ^, the cubic nonlinear coupling coefficients are evaluated. Furthermore, the total derivatives ^^ ^,^,^ ^^ of the cubic nonlinear coupling coefficients according to the design parameters ^ are determined. In a next step of the optimization loop, the cubic nonlinear coupling coefficients ^ evaluated for the determined eigenfrequencies ^ and eigenvectors ^ as well as the total derivatives ^^ ^,^,^ ^,^,^ n ^^ of the cubic nonlinear coupling coefficients for all design parameters ^ are passed to an optimization algorithm to optimize an objective function. The values ​​for the cubic nonlinear coupling coefficients ^ ^,^,^ and the total derivatives ^^ ^,^,^^^ of the cubic nonlinear coupling coefficients of the objective function or considered as constraints for the optimization algorithm. R. 410718 - 5 -For example, a constraint can be specified that the magnitude of one, several, or all cubic nonlinear coupling coefficients should be smaller than a specified value. By optimizing the objective function, new values ​​for each of the design parameters ^ can then be determined. In a subsequent step, new coordinates ^^(^) = ^^,^ + Δ^^(^) are applied at least for the subset of nodes by applying the shifts Δ^^(^) evaluated for the values ​​of the design parameters ^ to the coordinates ^^,^ of the nodes, thus shifting 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 ^ ^,^,^for the eigenfrequencies ^ and / or eigenvectors ^ are checked. The displacements Δ^^(^) of the coordinates can depend linearly on the design parameters. In principle, however, other functional dependencies of the displacements Δ^^(^) on the design parameters ^ are also possible. Furthermore, it can be provided that a specific design parameter ^ influences the coordinates of all nodes of the subset of nodes, or it can be provided that a design parameter ^ only influences the coordinates ^^,^ of one or a few nodes of the subset of nodes. 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. With further advantage, it can be provided that the total derivatives ^^ of the cubic nonlinear coupling coefficients with respect to the design parameters ^ can be determined by an adjoint sensitivity analysis. With even further advantage, the subset of nodes can be provided to include boundary nodes of the FEM model or to consist of boundary nodes of the FEM model. R. 410718 - 6 -This means, in particular, that only the edge nodes that discretize the edge of the sensor geometry are made dependent on the design parameters ^, or that only for the edge nodes are shifts Δ^^(^) of the coordinates defined as a function of the design parameters. Since, in particular, the inner nodes of the sensor geometry are not dependent on the design parameters ^, the computing time can be significantly reduced. Furthermore, it can be provided that the optimization loop includes 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 nodes, wherein the new coordinates are preferably determined using an elastic model of the sensor geometry, wherein more preferably the new coordinates of the nodes of the subset of nodes are considered as boundary conditions, in particular as Dirichlet boundary conditions.Once the new coordinates ^^(^) =^^,^ + Δ^^(^) for the subset of nodes, in particular for the edge nodes, of the FEM model have been determined within the optimization loop, the positions of the nodes not belonging to the subset, in particular the inner nodes, of the FEM model are adjusted. To track the nodes not belonging to the subset, in particular the inner nodes, an elastic model of the sensor geometry can be used. The tracking of the inner nodes can in particular be determined simply proportional to the displacements Δ^^(^) of the nodes belonging to the subset. It can advantageously be provided that the cubic nonlinear coupling coefficients are given by. (^, ^)^^ (1)where ^ is a design parameter, where ^ ^^⃗^,^ ^^⃗^ ,^ ^^⃗^ are eigenvectors of the eigenmodes n, m, l, where ^ is an index of the finite elements, where ^ is an R. 410718 -7 - Integration point index, where ^ ^ is a matrix containing material parameters for the finite element ^, where ^ and ^^^^, ^,^ ^^⃗ ^^^ and ^ (^, ^) are matrices for the finite element ^, which result from the FEM model of the sensor geometry. Furthermore, it can be provided that the cubic nonlinear coupling coefficients are normalized according to the following rule: ^^nn ^, ^, ^ are three different indices if an index appears twice in ^, ^, ^ (2) (5b) if ^ = ^ = ^In principle, any cubic nonlinear coupling coefficient ^ ^,^,^depend on all design parameters ^. Formula (1) specifies the explicit dependencies on the design parameters ^. Implicitly, the eigenvectors ^ also depend on the design parameters ^. In formula (1), the sum is calculated over all finite elements ^. The index on the matrices and vectors indicates that these must be evaluated individually for each finite element ^. For each finite element ^, the sum is calculated over the integration points, which are indexed by the integration point index ^. The material parameters of the matrices ^ ^ depend on the material, e.g. silicon, from which the finite element ^ is made. The matrices ^ und and ^ (^, ^) are matrices that are specific to each finite element ^ and also depend on the integration point ^ and the design parameters ^. The matrices also depend on the eigenvectors ^ ^of the eigenmodes. Formula (1) can be obtained as follows: The time-dependent displacement ^⃗ (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: R. 410718 - 8 - Here, t is the time and ^^ is 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: ^ = ∫^^(^)⃗: ^(^)⃗ ^^(4)The integrand is the element-wise scalar product of the matrices ^ and ^. ^ is the second Piola-Kirchhoff stress tensor, ^ the Green-Lagrange strain tensor, and ^ the volume of the sensor geometry. Furthermore, linear elastic material behavior is assumed. ^ and ^ then contain both linear and quadratic terms in ^.⃗ Substituting equation (3) into (4) and collecting the terms that are cubic in the eigenvectors ^ allows the cubic nonlinear coupling coefficient ^^,^,^ , which couples the modes n, m, and l, to be calculated according to (1).It is preferably provided that the optimization loop comprises the step of determining a mass matrix ^ and / or a stiffness matrix ^ of the sensor geometry, and that the natural frequencies ^ and eigenvectors ^ of the sensor geometry are determined from the mass matrix ^ and / or the stiffness matrix ^, in particular by solving an eigenvalue problem for the mass matrix ^ and / or the stiffness matrix ^. The natural frequencies ^ and eigenvectors ^ of the sensor geometry can be determined by solving the eigenvalue problem. where ^^ = 2^^^ with the natural frequency ^^ is the eigenmode i and where ^ ^ ^⃗ ^ is the eigenvector of eigenmode i. The mass matrix ^ and the stiffness matrix ^ 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 taken into account that they are mass-normalized: R. 410718 -9 - Preferably, the total derivatives of the cubic nonlinear coupling coefficients with respect to the design parameters ^ are given by where ^^⃗ ^ and ^^ are adjoint variables. The summation index ^ runs over the unique eigenmodes ^, ^, ^. If, for example, ^ = ^ then ^ = ^ and ^ = ^ are evaluated and summed. The derivatives of the system matrices ^ and ^ are calculated for each element ^ based on^^⃗ ^^ using where ^^ is the density of element ^ and ^ is a matrix resulting from the FEM discretization of the sensor geometry. ^ The derivative ^^ considers the derivative according to the explicit dependence on the design parameters ^ (in contrast, ^ ^^ as a total derivative also considers the implicit dependence via the eigenvectors ^, which also depend on the design parameters ^) and is calculated based on^^⃗ ^^ ^,^,^ ^^ calculated. ^^ is calculated according to: R. 410718 -10 - Afterwards just like ^ ^,^,^ in equation (2).^^⃗ ^ and ^^ are the adjoint variables. They must be for each eigenmode ^, of which^ ^,^,^ depends, are calculated once (regardless of the number of design parameters). ^ ^ is calculated according to: Depending on whether ^ = ^, ^ = ^ or ^ = ^: Here ^ ^ a matrix resulting from the connectivity of the FEM mesh and assigning the degrees of freedom of element ^ to the corresponding global degrees of freedom. R. 410718 - 11 - Subsequently, an additional normalization is carried out in the following cases: If ^ = ^ = ^ then ^^ ^,^,^ ^ ^ ^^^⃗ ^ → ^^ ^,^,^ ^ ^^^^⃗ ^ . If exactly two indices of ^, ^, ^ are identical, only for the third (the single) index ^^ ^,^,^ ^ ^^ ^,^,^ ^^^^⃗ ^ → ^ ^^^^⃗ ^.A global ^^⃗ ^ is then determined from the solution of the following system of equations: Then the adjoint variable for each element can be determined using ^^⃗ ^ ^^ = ^ ^^⃗ ^. Thus, equation (5) can be evaluated and the calculation of the total derivative ^^ ^,^,^ ^^ be completed. It can be further advantageously provided that the method further comprises the step of defining geometric boundary conditions for the sensor geometry, wherein the objective function is preferably also a function of the boundary conditions for the sensor geometry. By taking geometric boundary conditions into account, it can be ensured that the sensor geometry meets specified technical requirements. In particular, the geometric boundary conditions can include minimum widths ^ ( ^ )and / or minimum distances ^(^) of elements or areas of the sensor geometry. The boundary conditions, in particular the minimum widths ^ ( ^ ) and / or the minimum distances ^ ( ^ ) , depend on the coordinates of the node points and thus at least indirectly also on the design parameters ^. For stability reasons, it can be provided that certain elements, such as beams or springs, of the sensor geometry do not have a certain minimum width ^(^) R. 410718 - 12 -or must have a certain minimum distance ^(^) from other elements, such as beams or springs. During the optimization loop, it can be checked whether the geometric boundary conditions, in particular the minimum widths ^(^) and minimum distances ^(^), are met. If the geometric boundary conditions are not met, the values ​​of the design parameters ^ determined by the optimization algorithm can be discarded. To determine whether the boundary conditions are met, the corresponding parameters, such as the widths and distances of elements or areas of the sensor geometry, can be determined. In particular, it can be provided that the widths or distances of the elements are determined using a ray tracing method.In a ray tracing method, the intersection point of an inwardly or outwardly directed normal vector with the nearest line segment is determined. To accelerate the process of searching for the nearest line segment hit by the normal vector, a kd-tree can be used to significantly reduce the number of line segments to be searched for. Preferably, the design parameters ^ include 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 in a layer thickness of the sensor geometry, and / or bending of the sensor geometry, and / or local variations in etching losses of a manufacturing process for a sensor.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. The sensor's deflections can be caused by intrinsic stress in the material and can be addressed via a corresponding design parameter in the process for optimizing the sensor geometry (R. 410718). 13 -be taken into account. Etching losses are to be expected during the manufacturing process for the sensor to be manufactured later based on the calculated sensor geometry. Such etching losses can also be included as design parameters for optimizing the sensor geometry. With further advantage, the sensor geometry is a sensor geometry of a MEMS sensor, in particular a MEMS angular rate sensor. With even further advantage, the sensor geometry can be a 2D geometry. In principle, geometric nonlinearities of three-dimensional sensor geometries can be optimized using the process. Modern MEMS sensors, in particular modern MEMS angular rate sensors, are often manufactured in an etching process along one axis. Accordingly, the optimization of the geometric nonlinearities can also be carried out based on a two-dimensional FEM model, which is then extruded for the production of the sensor.A further solution to the problem underlying the invention consists in a method for producing a sensor, in particular a MEMS sensor, more particularly a MEMS yaw rate sensor, wherein a sensor geometry was created using 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, more preferably in an etching process. The invention is explained in more detail below with reference to the attached figure. The single figure shows a flowchart for a computer-implemented method for optimizing geometric nonlinearities of a sensor geometry. The figure explains a computer-implemented method 100 for optimizing geometric nonlinearities of a sensor geometry 10 in accordance with the invention.In a first process step S1, an initial FEM model 11 of a sensor geometry 10 is created. The initial FEM model 11 includes R. 410718 -. 14 - 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 ^ ^,^ assigned. In a second process step S2, design parameters ^ of the sensor geometry 10 are defined. In addition, displacements Δ^^(^) 15 of the coordinates ^^,^ of boundary nodes 16 are defined as a function of the design parameters ^. A design parameter ^ can be defined such that it shifts individual or multiple boundary nodes 16. In addition, for each design parameter ^, the partial derivatives^^^^ ^⃗ ^^ of the displacements Δ^^(^) 15 are determined. In addition, in process step S2, cubic nonlinear coupling coefficients ^^,^,^ = ^^,^,^(^, ^, ^) are represented as a function of the design parameters ^, of natural frequencies ^ and of natural vectors ^ to eigenmodes of the sensor geometry 10, whereby the cubic nonlinear coupling coefficients ^^,^,^ couple the equations of motion of the natural modes n, m, l of the sensor geometry 10 with each other. In a third process step S3, k natural frequencies ^ and natural vectors ^ to eigenmodes of the sensor geometry 10 are determined. Subsequently, in a further process step S4, the cubic nonlinear coupling coefficients ^^,^,^ = ^^,^,^(^, ^, ^) for the design parameters ^, the eigenfrequencies ^ and the eigenvectors ^ from step S3 are evaluated. In step S4, the total derivatives ^^ ^,^,^^^ of the cubic nonlinear coupling coefficients according to the design parameters ^ is determined. In step S5, a check is made to determine 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 ^ are determined that optimize the objective function. In a further method step S6, new coordinates ^^(^) = ^^,^ + Δ^^(^) are 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 process step S5, the process 100 is terminated in a final process step S9.

Claims

R. 410718 - 15 - Claims 1. Computer-implemented method (100) for optimizing geometric nonlinearities of a sensor geometry (10), comprising the steps of - creating an initial FEM model (11) of a sensor geometry (10), wherein the initial FEM model (11) comprises a plurality of finite elements ^ (12) which are formed from nodes (14) connected to line segments (13), wherein each node (14) is assigned a set of coordinates ^ ^,^ is assigned,- defining design parameters ^ of the sensor geometry (10),- defining displacements Δ^^(^) (15) of the coordinates ^^,^ of at least a subset of the node points (14) as functions of the design parameters ^,- determining the partial derivatives^^^^ ^ ⃗ ^^of the displacements Δ^^(^) (15) according to the design parameters,- Representation of cubic nonlinear coupling coefficients ^^,^,^ = as a function of at least the design parameters ^ and eigenvectors ^ to eigenmodes, and preferably eigenfrequencies ^, of the sensor geometry (10), where the cubic nonlinear coupling coefficients ^ ^,^,^ Coupling the equations of motion of the eigenmodes n, m, l of the sensor geometry (10), - Performing an optimization loop with the steps: o Determining k eigenfrequencies ^ and eigenvectors ^ of the sensor geometry (10), where k ≥ max {n, m, l}, o Determining values for the cubic nonlinear coupling coefficients^^,^,^ for the eigenfrequencies ^ and / or eigenvectors ^, o Determining total derivatives ^^ ^,^,^ ^^ of the cubic nonlinear coupling coefficients according to the design parameters ^,o Determining values of the design parameters ^ using an optimization algorithm, which optimizes a target function, where the values for the cubic nonlinear coupling coefficients ^ ^,^,^ and the total derivatives ^^^,^,^ ^^ the cubic nonlinear coupling coefficients of the objective function R. 410718 - 16 - and / or are taken into account as constraints when executing the optimization algorithm,o Determining new coordinates ^^(^) = ^^,^ + Δ^^(^) at least for the subset of the nodes (14) by applying the shifts Δ^^(^) evaluated for the values of the design parameters ^ to the coordinates ^^,^ of the nodes,- Repeating 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 ^^ ^,^,^^^ the cubic nonlinear coupling coefficients according to the design parameters ^ are 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 boundary node points (16) of the FEM model (11) or consists of boundary 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 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 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. R. 410718 - 17 - ^^,^,^ = ^^,^,^^^,^ ^^⃗^,^ ^^⃗^ ,^ ^^⃗^^= ^ ^ ^^^^^^⃗ ^ ^ ^ ^ ^ ^ ^ ^^ ^ ^ (^, ^)^ ^ ^ ^^^^, ^,^ ^^⃗^^^^^^⃗^^ ^ + ^^^^⃗ ^ ^ ^ ^ ^ ^ ^^ ^(^, ^)^ ^ ^ ^ ^ ^ ^^^, ^,^ ^^⃗^^^^^^⃗^ + ^^^^⃗ ^ ^ ^ ^ ^^ ^ ^^(^, ^)^ ^ ^ ^ ^ ^^^, ^,^ ^^⃗^^^^^⃗^ ^^ ^ ⋅ det^^ (^, ^)^^where ^ is a design parameter, where ^ ^^⃗^,^ ^^⃗^ ,^ ^^⃗^ are eigenvectors of the eigenmodes n,m,l, where ^ is an index of the finite elements (12), where ^ is an integration point index, where ^ ^ is a matrix containing material parameters for the finite element ^, where ^ ^^(^, ^) and ^ and ^ (^, ^) are matrices for the finite element ^ 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 ^ and / or a stiffness matrix ^ of the sensor geometry (10), and that the eigenfrequencies ^ and eigenvectors ^ of the sensor geometry (10) are determined from the mass matrix ^ and / or the stiffness matrix ^, in particular by solving an eigenvalue problem for the mass matrix ^ and / or the stiffness matrix ^.

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 ^ are determined by are given, where und 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 nonlinear coupling coefficients ^ ^,^,^ and / or reaching a predetermined threshold value of the target function and / or reaching a predetermined threshold value of a gradient of the target function.

10. Method for producing a sensor, in particular a MEMS sensor, further in particular a MEMS rotation rate sensor, wherein a R. 410718 - 18 - 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 manufactured on the basis of the optimized sensor geometry (10), preferably in an etching process.

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