Computer-implemented method for optimizing sensor geometries

Abstract

In order to be able to efficiently calculate sensor geometries for a plurality of design parameters, a computer-implemented method (100) is proposed comprising the steps of -generating an initial FEM model (11) of a sensor geometry (10), wherein the initial FEM model (11) comprises a plurality of node points i connected by line segments, each node point i being assigned a set of coordinates x i,0 ; - defining design parameters p k of the sensor geometry (10), -defining displacements Δx i (p k ) of the coordinates x i,0 of at least one subset of node points as functions of the design parameters p k ; defining a target function o = o(ƒ,Φ,p k ), said target function being a function of at least the design parameters p k , natural frequencies ƒ, and / or eigenvectors Φ for eigenmodes of the sensor geometry, - carrying out an optimization loop having the steps of ○ascertaining natural frequencies ƒ and eigenvectors Φ of the sensor geometry, ○ascertaining a total derivative [formula I] dpk of the target function o according to the design parameters p k , ○determining values of the design parameters p k by means of an optimization algorithm which optimizes the target function o, and ○determining new coordinates x i (p k ) = x i,0 + Δx i (p k ) at least for the subset of node points by applying the displacements Δx i (p k ) defined for the values of the design parameters p k to the coordinates x i,0 of the node points, and - repeating the optimization loop until a termination criterion is met.

Description

[0001] R. 410717 - 1 -Description Title Computer-implemented method for optimizing sensor geometries State of the art The present invention relates to a computer-implemented method for optimizing sensor geometries. In the development of sensors such as MEMS sensors, in particular MEMS yaw rate sensors, as well as sensors of many other sensor classes, finite element method models (FEM models) of the respective sensor geometries are created and analyzed with regard to their eigenmodes as part of a modal analysis. To optimize sensor geometries with regard to product-specific requirements, a series of design parameters such as the widths and lengths of elements of the sensor geometry are usually defined. A design that meets the desired requirements is then found using optimization methods.Such development processes are costly, as many design iterations are required, which require large computing resources even with the most modern evolutionary algorithms. Furthermore, since a new FEM model must be created for each set of design parameters during optimization, the computing time increases extremely rapidly. Disclosure of the Invention: The object of the present invention is to provide a computer-implemented method with which sensor geometries can be calculated for a variety of R. 410717 -. 2 -Design parameters can be calculated efficiently and a large parameter space can be analyzed effectively. To achieve the object underlying the invention, a computer-implemented method for optimizing sensor geometries is proposed, comprising the steps of: - Creating an initial FEM model of a sensor geometry, wherein the initial FEM model comprises a plurality of nodes connected by line segments, wherein each node i is assigned a set of coordinates ^^,^, - Defining design parameters ^^ of the sensor geometry, - Defining displacements Δ^^(^^) of the coordinates ^^,^ of at least a subset of the nodes as functions of the design parameters ^ ^,- Defining an objective function ^ = ^(^, ^, ^^), where the objective function is a function of at least the design parameters ^^, eigenfrequencies ^ and / or eigenvectors ^ to eigenmodes of the sensor geometry,- Performing an optimization loop with the steps of determining eigenfrequencies ^ and eigenvectors ^ of the sensor geometry, determining a total derivative^^ ^ ^ ^ the objective function ^ according to the design parameters ^ ^ , o Determining values ​​of the design parameters ^^, using an optimization algorithm that optimizes the objective function ^, o Determining new coordinates ^^(^^) = ^^,^ + Δ^^(^^) for at least the subset of 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 i 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 (^, ^)^,^ . R. 410717 - 3 - The initial FEM model is preferably created using known methods. For the design parameters ^ ^ In particular, these can 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 i as a function of the design parameters ^ ^ By applying the shifts Δ^^(^^) to the coordinate ^^,^ of the i-th node according to ^^(^^) = ^^,^ + Δ^^(^^), new coordinates are assigned to the i-th node assigned. The objective function ^ = ^(^, ^, ^^) defined according to the method represents an optimization goal, which is to be achieved by adapting or changing the sensor geometry. The eigenvectors ^ describe the eigenmodes or shrinkage modes of the sensor geometry. The computer-implemented method according to the invention further comprises an optimization loop, wherein within the optimization loop, the eigenfrequencies ^ and eigenvectors ^ of the eigenmodes of the sensor geometry are first determined. With the determined eigenfrequencies ^ and eigenvectors ^ as well as the design parameters ^^, the objective function can be evaluated for each run of the optimization loop. Furthermore, within the ^^ optimization loop, the total derivative ^^ ^ of the objective function with respect to the design parameters. The total derivative can also be referred to as the gradient of the objective function with respect to the design parameters. Using ^^ the total derivative ^^^ the objective function, the design parameters ^ ^ varied using the optimization algorithm to find values ​​of the design parameters^^ that optimize the objective function ^. An optimization of the objective function ^ can be understood as a maximization or minimization of the objective function. For the values ​​of the design parameters ^ determined in this way, ^ The coordinate shifts Δ^^(^^) are evaluated, and new coordinates are determined for the nodes according to ^^(^^) = ^^,^ +Δ^^(^^). The optimization loop is then repeated until a termination criterion is met. The termination criterion can, for example, be the achievement, in particular the R. 410717 - 4 - Exceeding or falling below a predetermined value of the objective function. The displacements Δ^^(^^) of the coordinates can depend linearly on the design parameters. However, other functional dependencies of the displacements Δ^^(^^) on the design parameters are also possible.^ possible. Furthermore, it can be provided that a certain design parameter ^ ^ the coordinates ^ ^,^ all nodes of the subset of nodes, or it may be provided that a design parameter ^ ^ only the coordinates ^ ^,^ one or more nodes of the subset of nodes. In the method according to the invention, variations in the sensor geometry are iteratively analyzed and optimized for a given design goal. With the method according to the invention, it is no longer necessary for each set of design parameters ^ ^ to create a new FEM model. Accordingly, the computation time can be significantly reduced. Compared to existing methods, the proposed method can produce optimized sensor geometries after just a few iteration steps of the order of 10 1 In addition, sensor geometries for several hundred design parameters ^^ be optimized. Furthermore, a full-shape optimization of sensor geometries is possible. The optimization algorithm is preferably a gradient-based optimization algorithm, particularly preferably a sequential quadratic programming (SQP) algorithm or a method of moving asymptotes (MMA) algorithm. The set of design parameters found by the optimization algorithm ^ ^ can concern a global or a local maximum or minimum of the objective function. R. 410717 - 5 - With further advantage, the subset of nodes can be provided to include edge nodes of the FEM model or to consist of edge nodes of the FEM model. This means, in particular, that only the edge nodes that discretize the edge of the sensor geometry are influenced by the design parameters ^ ^are made dependent, or that only for the edge nodes, displacements Δ^^(^^) of the coordinates are 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. Preferably, it can further 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 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 nodes are used as boundary conditions, in particular as Dirichlet boundary conditions.After the new coordinates ^^(^^) = ^^,^ + Δ^^(^^) for the subset of nodes, particularly 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, particularly 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, particularly the inner nodes. The tracking of the inner nodes can, in particular, be determined simply proportional to the displacements Δ^^(^^) of the nodes belonging to the subset. R. 410717 -. 6 -With further advantage, it can be 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 regions of the sensor geometry. The boundary conditions, in particular the minimum widths ^(^^) and / or the minimum distances ^(^ ^ ), depend on the coordinates of the nodes and thus at least indirectly on the design parameters ^ ^For stability reasons, it can be stipulated that certain elements of the sensor geometry, such as beams or springs, must not be less than a certain minimum width^(^^) or must have a certain minimum distance ^(^^) to other elements, such as beams or springs. During the optimization loop, it can thus 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 ^ ^be discarded. To determine whether the boundary conditions are met, the corresponding parameters, such as the widths and spacing of elements or areas of the sensor geometries, can be determined. In particular, it can be provided that the widths or spacing of the elements are determined using a ray tracing method. R. 410717 - 7 - 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 that is hit by the normal vector, a kd-tree can be used to significantly reduce the number of line segments to be searched. ^^ Preferably, the determination of the total derivative ^^ ^ the objective function ^ according to the design parameters ^^ determining the partial^^ ^^ ^^ derivatives ^^^ , ^^ and / or ^^the objective function according to the design parameters ^ ^ ,the eigenfrequencies ^ and / or the eigenvectors ^.The functional form of the partial derivatives The objective function with respect to the design parameters ^^, the eigenfrequencies ^ and / or the eigenvectors ^ can be determined before the first run through the optimization loop. The partial derivatives can be determined from the objective function analytically or numerically, or by means of an automatic differentiation method. A further advantage is that the determination of the total derivative ^^ ^^ ^ of the objective function according to the design parameters, determining the partial derivatives ^ ^ ^ , ^^ ^ the eigenfrequencies ^ and / or the eigenvectors ^ according to the design parameters ^ ^ comprises, wherein preferably the partial derivatives ^^ ^^ ^^ ^ , ^^^ the natural frequencies ^ and / or the eigenvectors ^ according to the design parameters ^ ^ be determined using an analytical method or a numerical method, more preferably an adjoint-state method. Since the eigenfrequencies ^ and eigenvectors ^ also depend on the design parameters ^ ^ may depend on the design parameters, the partial derivatives of the eigenfrequencies and / or eigenmodes are preferably also determined in the method. R. 410717 - 8 -It is further advantageous 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 eigenfrequency ^^ of mode i, and where ^ ^^⃗^is the eigenvector of mode i. The mass matrix ^ and the contention 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: Partial derivatives are preferred ^ ^ ^ the natural frequencies ^ according to the design parameters ^ ^ according to the formula The partial derivatives ^ ^ ^ of the eigenvectors ^ according to the design parameters ^ ^ can be determined using an analytical method or a numerical method, preferably an adjoint-state method. R. 410717 - 9 - With further advantage, it can be provided that the termination criterion is reaching a threshold value of the objective function and / or a threshold value of the ^^ total derivative ^^ ^ of the objective function. ^^ In particular, if the total derivative ^^ ^of the objective function, ie the gradient of the objective function, falls below a threshold value close to 0, the termination criterion can be met, since then the set of design parameters corresponds to a value of the objective function close to a global or local minimum or maximum. Preferably, the design parameters ^ ^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 bending of the sensor can be caused by intrinsic stress in the material and can be taken into account in the process for optimizing the sensor geometry via a corresponding design parameter.For the sensor to be manufactured based on the optimized sensor geometry, etching losses are to be expected during the manufacturing process. Such etching losses can also be included as design parameters for optimizing the sensor geometry. It can further be advantageous for the objective function to describe an optimization goal, wherein the optimization goal is an optimization of the natural frequencies, and / or the frequency spacings of the natural frequencies from one another or to multiples of a drive frequency, of free spaces of frequency bands, and / or an optimization of the natural modes, and / or a reduction of mechanical stress in an natural mode, and / or a R. 410717 -. 10 -This includes inducing mode shapes, and / or preventing false signals due to external excitations, and / or increasing the freedom of movement of an eigenmode, and / or optimizing a mechanical quality, and / or optimizing a target parameter of a sensor. The optimization goal of clearing individual frequency bands is particularly advantageous when excitation during the manufacturing process, for example, during wire bonding, or during operation of the sensor could lead to unwanted false signals or damage to the sensor. To avoid breakage, especially during drive movements, an optimization goal can also be the reduction of mechanical stresses in the eigenmodes. Furthermore, it can be useful to induce specific mode shapes as an optimization goal.If the optimization goal is to prevent false signals due to external excitation, this can be achieved, for example, by balancing individual motion patterns, which can also include multiple modes. Furthermore, the optimization goal can be to optimize the scattering of all of the aforementioned variables during the manufacturing process. It is further advantageous for the sensor geometry to be a sensor geometry of a MEMS sensor, in particular a MEMS angular rate sensor. It is even further advantageous for the sensor geometry to be a 2D geometry. In principle, three-dimensional sensor geometries can be optimized using the method. Modern MEMS sensors, in particular modern MEMS angular rate sensors, are often manufactured in an etching process along an axis. Accordingly, the optimization of the sensor geometry can also be based on a two-dimensional FEM model, which is then extruded for the production of the sensor.410717 -. 11 -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 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 figures. They show: Fig. 1 a flow chart of a computer-implemented method for optimizing sensor geometries, Fig. 2a a first sub-area of ​​an FEM model of a sensor geometry, Fig. 2b a second sub-area of ​​an FEM model of a sensor geometry, and Fig. 2c a third sub-area of ​​an FEM model of a sensor geometry. Fig.1 shows a flowchart of a computer-implemented method 100 for optimizing sensor geometries 10 in accordance with the invention. Figs. 2a to 2c show partial regions of sensor geometries 10. In a first method step S1, an FEM model 11 of a sensor geometry 10 is created. In a second method step S2, design parameters ^^ and displacements Δ^^(^^) of the coordinates ^^,^ of boundary nodes 12 of the sensor geometry 10 are defined as a function of the design parameters ^^, and an objective function ^ = ^(^, ^, ^^) is defined as a function of the design parameters ^^, natural frequencies ^, and eigenvectors ^ for eigenmodes of the sensor geometry 10. An optimization loop follows method step S2. In a method step S3, natural frequencies ^ and eigenvectors ^ of the sensor geometry 10 are determined. In method step S4, a total derivative ^^ is calculated. ^^^ the objective function ^ according to the design parameters ^^ and then R. 410717 - 12 - In a process step S5, values ​​of the design parameters ^^ are determined using an optimization algorithm, which optimize the objective function ^. Subsequently, in step S6, new coordinates ^^(^^) = ^^,^ + Δ^^(^^) for the boundary nodes 12 are determined by applying the displacements Δ^^(^^) evaluated for the values ​​of the design parameters ^^ to the coordinates ^ ^,^determined. In method step S7, new coordinates for the inner nodes 13 of the FEM model 11 are determined using an elastic model of the sensor geometry 10. In method step S8, it is checked whether a termination criterion is met. The termination criterion can, for example, be reaching a threshold value of the objective function ^. If the termination criterion is met, the method 100 ends with method step S9. Otherwise, the method 100 is repeated starting with method step S3. Using the partial areas of sensor geometries 10 shown in Figs. 2a to 2c, various design parameters ^ are determined as examples. ^ explained. According to Fig. 2a, a design parameter ^ ^ a width 14 of an element 15 of the sensor geometry 10. According to Fig.2b, etching losses 16 expected during the manufacture of a sensor can also be determined by design parameters ^ ^be taken into account. As also shown in Fig. 2c, variations17 of individual edge nodes 12 can also be included in the optimization of the sensor geometry 10 by design parameters ^^.

Claims

R. 410717 - 13 - Claims 1. Computer-implemented method (100) for optimizing sensor geometries 10, comprising the steps of - creating an initial FEM model of a sensor geometry, wherein the initial FEM model comprises a plurality of nodes i connected by line segments, wherein each node i 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 nodes as functions of the design parameters ^ ^ ,- Define an objective function ^ = ^(^, ^, ^^), where the objective function is a function at least dependent on the design parameters ^ ^ , eigenfrequencies ^ and / or eigenvectors ^ to eigenmodes of the sensor geometry,- Performing an optimization loop with the stepso Determining eigenfrequencies ^ and eigenvectors ^ of the sensor geometry,o Determining a total derivative^^ ^ ^^ the objective function ^ according to the design parameters ^ ^ ,o Determining values of the design parameters ^^, using an optimization algorithm which optimizes the objective function ^,o Determining new coordinates ^^(^^) = ^^,^ + Δ^^(^^) for at least the subset of the nodes by applying the displacements Δ^^(^^) evaluated for the values of the design parameters ^^ to the coordinates ^ ^,^ the nodes,- repeating the optimization loop until a termination criterion is met.

2. The computer-implemented method (100) according to claim 1, wherein the optimization algorithm is a gradient-based optimization algorithm, preferably a sequential quadratic programming (SQP) algorithm or a method of moving asymptotes (MMA) algorithm. R. 410717 - 14 -3. The computer-implemented method (100) according to claim 1 or 2, wherein the subset of nodes comprises edge nodes (12) of the FEM model (11) or consists of edge nodes (12) of the FEM model (11).

4. The computer-implemented method 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 of the FEM model (11) that do not belong to the subset of nodes, wherein the new coordinates are preferably determined using an elastic model of the sensor geometry (10), wherein more preferably the new coordinates of the nodes of the subset of nodes are used as boundary conditions, in particular as Dirichlet boundary conditions. 5.Computer-implemented method according to one of the preceding claims, wherein the method further comprises the step of defining geometric boundary conditions for the sensor geometry (10), wherein the objective function ^ is preferably further a function of the boundary conditions for the sensor geometry (10), wherein the geometric boundary conditions preferably comprise minimum widths ^(^^) and / or minimum distances ^(^^) of elements (15) or regions of the sensor geometry (10).

6. Computer-implemented method (100) according to one of the preceding^^ claims, wherein determining the total derivative. ^^ ^ the objective function ^ according to the design parameters ^ ^ determining the partial derivatives ^^ ^ , ^^ ^^ and / or ^^the objective function according to the design parameters ^ ^, the eigenfrequencies ^ and / or the eigenvectors ^ 7. Computer-implemented method (100) according to one of the preceding^^ claims, wherein determining the total derivative ^^ ^ of the objective function according to the design parameters, determining the partial derivatives ^ ^ ^ , ^^ ^ the eigenfrequencies ^ and / or the eigenvectors ^ according to the design parameters ^ ^ wherein preferably the partial derivatives R. 410717 - 15 - ^ ^ ^ , ^^ ^ the natural frequencies ^ and / or the eigenvectors ^ according to the design parameters ^ ^with an analytical method or a numerical method, more preferably an adjoint-state method.

8. 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 ^.

9. Computer-implemented method (100) according to one of the preceding claims, wherein the termination criterion is reaching a threshold value of the objective function ^ and / or a threshold value of the total^^ derivative ^^ ^the objective function is.

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 sensor geometries 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.

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