Method for establishing microstructure surrogate model, and application and device thereof
By using orthogonal least squares to filter sample points in the microstructure surrogate model, a new surrogate model is constructed, which solves the problems of low computational efficiency and wasted storage space caused by too many sample points in the existing technology, and achieves efficient computation and storage optimization.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- HUAZHONG UNIV OF SCI & TECH
- Filing Date
- 2023-12-08
- Publication Date
- 2026-06-09
AI Technical Summary
Existing technologies require a large number of sample points to ensure accuracy when constructing microstructure proxy models, but this leads to a waste of computational efficiency and storage space, and cannot effectively shorten the solution time.
The orthogonal least squares method is used to select the sample points that have the greatest impact on the output from a large number of input sample points, and a new surrogate model is constructed. By selecting the sample points with the largest error and processing them with orthogonal feature vectors, the number of sample points is reduced and the model size is reduced.
While maintaining accuracy, it significantly improves computational efficiency and reduces storage space usage, while also providing greater efficiency for subsequent iterations to reduce the model size.
Smart Images
Figure CN117634307B_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the technical field of structural optimization design, and more specifically, relates to a method for establishing a microstructure proxy model and its application and device. Background Technology
[0002] Dual-scale structures possess advantages such as light weight, high strength, and vibration reduction, and have great application potential. A dual-scale structure optimization method based on the level set approach can obtain well-optimized structures while ensuring the continuity of the microstructure.
[0003] In the level set method for dual-scale structural optimization design, a microstructure model needs to be constructed. Due to the complexity of computational homogenization algorithms, directly solving the microstructure model requires a significant amount of time. Using the original model is not feasible in optimization algorithms where a single step requires thousands of solutions. Therefore, using surrogate models to simulate high-precision models is a common method in engineering, such as radial basis functions (RBF), neural networks, response surface models, and Kriging models. RBF maps complex design spaces through weighted summations and interpolations of simple functions, has no requirements on response characteristics, and can fit any type of function well. The performance of the surrogate model is highly dependent on the distribution and number of sample points in the modeling space. A suitable number of evenly distributed sample points can effectively reflect the spatial variation trends and information of the real model, ensuring high accuracy and reliability of the information provided when using the surrogate model. However, in practical engineering, it is often impossible to know the appropriate number of sample points. To ensure accuracy, an excessive number of sample points are often chosen, which undoubtedly reduces computational efficiency. Summary of the Invention
[0004] To address the above-mentioned deficiencies or improvement needs of existing technologies, this invention provides a method for establishing a microstructure surrogate model, its application, and a device. The method selects M (M << N) sample points from the N input sample points of the initial surrogate model that have the most significant impact on the output and assembles them into a new surrogate model, thereby shortening the solution time.
[0005] To achieve the above objectives, according to one aspect of the present invention, a method for establishing a microstructure proxy model is provided, the method comprising the following steps:
[0006] (1) The elastic matrix and volume fraction of the combined microstructures corresponding to different cutting heights are used to form a sample point database, and then each cutting height in the sample point database is used as the input center point of the radial basis function interpolation to construct an initial surrogate model.
[0007] (2) The orthogonal least squares method is used to select each sample point in the sample point database in turn and calculate the error it brings. Then, the sample point corresponding to the maximum error is selected as the first sample point of the final proxy model.
[0008] (3) Select the remaining sample points in the sample point database in sequence, orthogonalize their corresponding feature vectors with the feature vectors of the selected final agent model sample points, calculate the error they bring, and then select the sample point corresponding to the maximum error in this round as the next sample point of the final agent model. Repeat the selection until the sum of the errors of the selected final agent model sample points is greater than or equal to the threshold.
[0009] (4) Calculate the weights of the sample points of the selected final agent model, and then construct the final agent model based on the sample points of the selected final agent model and their corresponding weights.
[0010] Furthermore, the unit is given a combined microstructure obtained by cutting with multiple independent level set functions and corresponding cutting heights, and the elasticity matrix of the combined microstructure at the two cutting heights is calculated by homogenization method.
[0011] Furthermore, for the final selected M sample points, the inverse solution matrix is assembled:
[0012]
[0013]
[0014] according to The least squares solution with weights:
[0015]
[0016] From the selected set of sample points With weight coefficients [α1, α2, α] M ] T Assemble the final agent model, which is represented as follows:
[0017]
[0018] Furthermore, the microstructure type is selected, and the level set function describing the microstructure is obtained; the level set function Φ is then used according to the level set function method. i (x) and the cutting function Ψ i (x) represents the combined function obtained from a single cutting operation.
[0019] γ i (x)=Φ i (x)-Ψ i (x), x∈D, i=1…l;
[0020] Where D is the reference domain, Ψ i (x)=h i, l represents the number of microstructure types, h i Let be the cutting height of the i-th level set function.
[0021] Furthermore, virtual microstructures are defined based on the properties of level set functions.
[0022]
[0023]
[0024]
[0025] The microstructures corresponding to multiple level set functions are obtained by merging them using a union Boolean operation:
[0026] γ k =min{γ1, γ2, ... γ l}
[0027] After merging, multiple microstructures within the reference domain are combined into a single final microstructure:
[0028]
[0029] The macroscopic equivalent elastic tensor of the obtained periodic microstructure Ω was obtained by calculating the homogenization theory:
[0030]
[0031] It can be represented in matrix form:
[0032]
[0033] The volume fraction of microstructure is obtained from the structural relationship:
[0034]
[0035] Where E pqrs For the local elastic tensor of the microstructure, Given the known macroscopic strain field in the horizontal direction. Given the known local strain field in the horizontal direction, The known macroscopic strain field in the vertical direction. Given the known local strain field in the vertical direction, DH represents the elastic tensor matrix expression of the microstructure with a cutting height of H. These are the nine elements in the elastic tensor, where V is the volume fraction, Ω is the size of the microstructure, and D represents the size of the entire reference domain.
[0036] Furthermore, a series of equidistant cutting heights H = [h1, h2, ..., h] are selected. lThe equivalent elasticity matrix of the corresponding N sets of reference microstructures is obtained:
[0037]
[0038] The initial surrogate model for radial basis functions is represented as follows:
[0039] [f] = [A][α] + [e]
[0040] in:
[0041]
[0042] α = [α1, α2, ..., α] N ] T
[0043] φ(||c i -c j ||) is a radial basis function, c = H = [h1, h2, ..., h l ] represents the input sample points, α N The weight coefficients corresponding to the Nth sample point;
[0044] Calculate P1=[φ(||c1-c1||),...,φ(||c N -c1||)] T Let c represent the feature vector of sample point c1. [A] is written as:
[0045] [A] = [P1, P2, ..., P] i ].
[0046] Further, the remaining sample points in the sample point database are selected sequentially, and their corresponding feature vectors are subjected to Schmitt orthogonalization with the feature vectors of the selected final proxy model sample points:
[0047]
[0048]
[0049] Where k≥1, represents the number of selected sample points, w j Let j be the orthogonal eigenvector of the j-th selected sample point; calculate the error introduced by the next sample point:
[0050]
[0051]
[0052] choose The largest sample point i is used as the next sample point in the final proxy model:
[0053]
[0054]
[0055]
[0056]
[0057] Calculate the total error of the selected sample points:
[0058]
[0059] The criteria for terminating the selection of sample points are as follows:
[0060] 1-err total ≤err set
[0061] Determine err total If the requirements are met, the selection of sample points ends; otherwise, S3 is repeated to select the (k+2)th sample point.
[0062] The present invention also provides a structural optimization method, wherein the optimization method constructs a proxy model of the structure to be optimized using the microstructure proxy model establishment method described above, and then uses the proxy model to perform structural optimization.
[0063] The present invention also provides a system for establishing a microstructure proxy model, the system including a memory and a processor, the memory storing a computer program, and the processor executing the computer program to perform the method for establishing a microstructure proxy model as described above.
[0064] The present invention also provides a computer-readable storage medium storing machine-executable instructions, which, when invoked and executed by a processor, cause the processor to implement the microstructure proxy model establishment method or the structure optimization method as described above.
[0065] In summary, compared with the prior art, the method for establishing a microstructure proxy model and its application and device provided by the present invention have the following beneficial effects:
[0066] 1. This invention uses orthogonal least squares to select the most useful sample points for the output from a large number of input sample points, thereby reducing the number of sample points and the size of the surrogate model. This allows the final surrogate model to greatly improve computational efficiency while ensuring accuracy.
[0067] 2. This invention uses the orthogonal least squares method to select the most useful sample points for the output from a large number of input sample points, thereby reducing the number of sample points, reducing the size of the proxy model, and thus reducing the storage space occupied by the device.
[0068] 3. In the process of selecting sample points in an orderly manner, the importance of the sample points is ranked. If it is desired to further reduce the scale, the previous points can be directly taken from the final proxy model for iteration, which is more efficient when the size of the proxy model is limited. Attached Figure Description
[0069] Figure 1 This is a flowchart illustrating a method for establishing a microstructure proxy model provided by the present invention;
[0070] Figure 2 In one embodiment of the present invention, the selected microstructure horizontal set function and the corresponding microstructure model when the cutting height is 0 are respectively a square horizontal set function, an X-shaped horizontal set function, a square microstructure and an X-shaped microstructure;
[0071] Figure 3 This is a schematic diagram of the distribution of sample points selected by the least squares method in an embodiment of the present invention;
[0072] Figure 4 These are fitting effect diagrams of the initial proxy model and the final proxy model obtained in an embodiment of the present invention, which are the initial proxy model and the final proxy model, respectively.
[0073] Figure 5 This is a fitting comparison diagram of the initial proxy model and the final proxy model obtained in an embodiment of the present invention with a fixed cutting height h1. Detailed Implementation
[0074] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.
[0075] This invention provides a method for establishing a microstructure proxy model, the method mainly including the following steps:
[0076] S1, the unit is given a combined microstructure obtained by cutting with multiple independent level set functions and corresponding cutting heights, and the elasticity matrix of the combined microstructure under two cutting heights is calculated by homogenization method. The different elastic matrices corresponding to different cutting heights and the volume fraction are used to form a sample point database. Then, each cutting height in the sample point database is used as the input center point of radial basis function interpolation to construct the initial surrogate model.
[0077] In this embodiment, a microstructure type is selected, and a level set function describing the microstructure is obtained. The level set function Φ is then used according to the level set function method. i (x) and the cutting function Ψ i (x) represents the combined function obtained from a single cutting operation.
[0078] γ i (x)=Φ i (x)-Ψ i (x), x∈D, i=1...l;
[0079] Where D is the reference domain, Ψ i (x)=h i , l represents the number of microstructure types, h i Let be the cutting height of the i-th level set function.
[0080] Virtual microstructures are defined based on the properties of level set functions.
[0081]
[0082]
[0083]
[0084] The microstructures corresponding to multiple level set functions are obtained by merging them using a union Boolean operation:
[0085] γ k =min{γ1, γ2, ... γ l}
[0086] After merging, multiple microstructures within the reference domain are combined into a single final microstructure:
[0087]
[0088] The macroscopic equivalent elastic tensor of the obtained periodic microstructure Ω was obtained by calculating the homogenization theory:
[0089]
[0090] It can also be represented in matrix form:
[0091]
[0092] The volume fraction of microstructure can be obtained from the structural relationship:
[0093]
[0094] Where E pqrs For the local elastic tensor of the microstructure, Given the known macroscopic strain field in the horizontal direction. Given the known local strain field in the horizontal direction, The known macroscopic strain field in the vertical direction. Given the known local strain field in the vertical direction, DH represents the elastic tensor matrix expression of the microstructure with a cutting height of H. These are the nine elements in the elastic tensor, where V is the volume fraction, Ω is the size of the microstructure, and D represents the size of the entire reference domain.
[0095] Using the above calculation and homogenization method, a series of equidistant cutting heights H = [h1, h2, ..., h] are selected. l The equivalent elasticity matrix of the corresponding N sets of reference microstructures is obtained:
[0096]
[0097] Use N sets of corresponding data as the input sample point database for the initial proxy model.
[0098] The initial surrogate model for radial basis functions is represented as follows:
[0099] [f] = [A][α] + [e]
[0100] in:
[0101]
[0102] α = [α1, α2, ..., α] N ] T
[0103] φ(||c i -c j ||) is a radial basis function, c = H = [h1, h2, ..., h l ] represents the input sample points, α N is the weight coefficient corresponding to the Nth sample point.
[0104] Calculate P1=[φ(||c1-c1||),...,φ(||c N -c1||)] TLet [A] represent the feature vector of sample point c1. [A] can be written as:
[0105] [A] = [P1, P2, ..., P] i ]
[0106] S2, using the orthogonal least squares method, each sample point in the sample point database is selected sequentially and the error it brings is calculated. Then, the sample point corresponding to the maximum error is selected as the first sample point of the final proxy model.
[0107] N sets of H vectors are used as the center point input to the radial basis function neural network. This is achieved by selecting N candidate sample points c = {c1, ..., c2}. N Select M sample points from} To reduce the size of the neural network.
[0108] The radial basis function surrogate model based on the least squares method is expressed as follows:
[0109]
[0110] in
[0111]
[0112]
[0113] For matrix Perform a Schmidt orthogonal transformation:
[0114]
[0115] Where W is a sequence of orthogonal columns w j The matrix such that U is an upper triangular matrix; combining the upper triangular matrix with the weight coefficients rewrites the interpolation model as follows:
[0116] [f] = [W][g] + [e]
[0117] in:
[0118]
[0119] Its least squares solution is:
[0120]
[0121] Because of w j They are orthogonal, and e is perpendicular to W. The sum of squares of the model yields:
[0122]
[0123] in It is the portion of the output error provided by each center point. It is the error provided by the candidate center point j.
[0124]
[0125] err j This represents the error compression ratio caused by sample point j.
[0126] Select P in the initial agent model in sequence i As the first sample point, let calculate Select the sample point with the largest error as the first sample point in the final proxy model:
[0127]
[0128]
[0129]
[0130] S3, sequentially select the remaining sample points in the sample point database, orthogonalize their corresponding feature vectors with the feature vectors of the selected final agent model sample points, calculate the resulting error, and then select the sample point corresponding to the maximum error in this round as the next sample point of the final agent model. Repeat the selection until the sum of the errors of the selected final agent model sample points is greater than or equal to the threshold.
[0131] From the remaining sample points, select sample points sequentially as the next sample points in the final surrogate model, and perform Schmitt orthogonalization on the orthogonal feature vectors of the selected sample points:
[0132]
[0133]
[0134] Where k≥1, represents the number of selected sample points, w j Let be the orthogonal feature vector of the j-th selected sample point.
[0135] Calculate the error introduced by the next sample point:
[0136]
[0137]
[0138] Selected The largest sample point i is used as the next sample point in the final proxy model:
[0139]
[0140]
[0141]
[0142]
[0143] Calculate the total error of the selected sample points:
[0144]
[0145] The criteria for terminating the selection of sample points are as follows:
[0146] 1-err total ≤err set
[0147] Determine err total If the requirements are met, the selection of sample points ends; otherwise, S3 is repeated to select the (k+2)th sample point.
[0148] S4, calculate the weights of the sample points of the selected final proxy model, and then construct the final proxy model based on the sample points of the selected final proxy model and their corresponding weights.
[0149] For the final selected M sample points, assemble the inverse solution matrix:
[0150]
[0151]
[0152] according to The least squares solution for the weights is known to be:
[0153]
[0154] From the selected set of sample points With weight coefficients [α1, α2, α] M ] T The final agent model is assembled as follows:
[0155]
[0156] The present invention also provides a structural optimization method, wherein the optimization method constructs a proxy model of the structure to be optimized using the microstructure proxy model establishment method described above, and then uses the proxy model to perform structural optimization.
[0157] The present invention also provides a system for establishing a microstructure proxy model, the system including a memory and a processor, the memory storing a computer program, and the processor executing the computer program to perform the microstructure proxy model establishment method as described above.
[0158] The present invention also provides a computer-readable storage medium storing machine-executable instructions, which, when invoked and executed by a processor, cause the processor to implement the microstructure proxy model establishment method or structural optimization method as described above.
[0159] The present invention will be further described in detail below with reference to specific embodiments.
[0160] Please see Figure 1 and Figure 2 In the elasticity coefficient matrix, D ij These models do not interfere with each other and, like V, belong to different proxy models. Furthermore, the construction methods for each proxy model are completely identical. Therefore, this implementation case selects D from the elasticity coefficient matrix. 11 The process of building the proxy model is used as an illustration.
[0161] This embodiment provides a method for establishing a microstructure surrogate model based on orthogonal least squares radial basis functions, including the following steps:
[0162] S1: Choose between square and X-shaped microstructures. When the horizontal set function of the microstructure and the corresponding cutting height are both 0, the shape of the microstructure is as follows: Figure 2 .
[0163] The true microstructure Ω of the unit is obtained by taking the union of the minimum values of the two microstructures. Then, a homogenization method is used to obtain 41×41 sets of corresponding data for each microstructure, with cutting heights ranging from -1 to 1 and intervals of 0.05. The database includes the equivalent elastic modulus D. H and equivalent volume fraction V H As an example, only D is taken here. 11 To ensure the accuracy of the surrogate model calculation in the boundary region, the database size was expanded from [-1, 1] to [-1.15, 1.15] using linear interpolation. Therefore, the final input sample points contain 2209 sets of data (47×47). The final database consists of the sample point coordinates and their corresponding equivalent elastic modulus D. H and equivalent volume fraction V H Composition, represented as follows:
[0164]
[0165] S2: The performance of the radial basis function surrogate model generally depends only on the selection of sample points and not on the selection of the radial basis function. Here, a multivariate quadratic function is chosen as the kernel function of the radial basis function. That is:
[0166]
[0167] Where ||c i -c j || represents the normal distance between sample points, and c0 is a small constant, which is taken as c0 = 0.05 here.
[0168] calculate
[0169]
[0170] S3: Import the initial proxy model and select the first sample point of the final proxy model. Then, sequentially import the c samples from S2... i As the first sample point, As the orthogonal feature vector of the first sample point, calculate And select the largest sample point as the first sample point.
[0171] S4: Select all remaining sample points for the final surrogate model. For the k selected sample points, sequentially select each sample point from the remaining Nk sample points as the next sample point for the final surrogate model, and perform Schmitt orthogonalization on the orthogonal feature vectors of the selected sample points:
[0172]
[0173]
[0174] Calculate the error introduced by the next sample point:
[0175]
[0176]
[0177] choose The largest sample point i is used as the next sample point in the final proxy model:
[0178]
[0179]
[0180]
[0181]
[0182]
[0183] Calculate the total error of the selected sample points:
[0184]
[0185] The criteria for terminating the selection of sample points are as follows:
[0186] 1-err total ≤err set
[0187] Here we retrieve err set =10 -6 , determine err total If the requirements are met, the selection of sample points ends; otherwise, S3 is repeated to select the (k+2)th sample point. In this example, 131 sample points were ultimately selected as the sample points for the final surrogate model, distributed as follows: Figure 3 As shown.
[0188] S5: Establish the final proxy model. For the 131 sample points finally selected, assemble the inverse matrix according to the S3-S4 process:
[0189]
[0190]
[0191] According to S3 The least squares solution for the weights is known to be:
[0192]
[0193] From the selected set of sample points With weight coefficients [α1, α2, α] 131 ] T Assemble the final proxy model, for any required cutting height H = [h1, h2], the elasticity matrix D corresponds to... 11 ,have:
[0194]
[0195] At this point, the elasticity matrix D is... 11 The size of the surrogate model was reduced from 2209 sample points to 131 sample points. Combined with the surrogate model for other remaining parameters, the average time for solving the single step in structural optimization was reduced from 21.768s to 1.674s, showing a significant improvement in efficiency.
[0196] The fitting results of the initial proxy model and the final proxy model are as follows: Figure 4 As shown, specifically, when the cutting height h1 = -0.6, the D in the initial proxy model and the final proxy model... 11 As h2 changes as Figure 5 As shown, the two curves almost overlap, indicating that although the number of sample points has been greatly reduced, the accuracy of the surrogate model has been almost unaffected.
[0197] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.
Claims
1. A method for establishing a microstructure proxy model, characterized in that, The method includes the following steps: (1) The elastic matrix and volume fraction of the combined microstructures corresponding to different cutting heights are used to form a sample point database, and then each cutting height in the sample point database is used as the input center point of the radial basis function interpolation to construct the initial surrogate model; (2) The orthogonal least squares method is used to select each sample point in the sample point database in turn and calculate the error it brings. Then, the sample point corresponding to the maximum error is selected as the first sample point of the final proxy model. (3) Select the remaining sample points in the sample point database in sequence, orthogonalize their corresponding feature vectors with the feature vectors of the selected final agent model sample points, calculate the error they bring, and then select the sample point corresponding to the maximum error in this round as the next sample point of the final agent model. Repeat the selection until the sum of the errors of the selected final agent model sample points is greater than or equal to the threshold. (4) Calculate the weights of the sample points of the selected final agent model, and then construct the final agent model based on the sample points of the selected final agent model and their corresponding weights; The unit is given a combined microstructure obtained by cutting with multiple independent level set functions and corresponding cutting heights, and the elasticity matrix of the combined microstructure under two cutting heights is calculated by homogenization method. For the final selected M sample points, assemble the inverse solution matrix: according to The least squares solution with weights: From the selected set of sample points with weight coefficients Assemble the final agent model, which is represented as follows: ； Select the microstructure type and obtain the level set function describing the microstructure; apply the level set function method. and cutting function This represents the combined function obtained from a single cutting operation. : Where D is the reference domain. Indicates the number of microstructure types. For the first i The cutting height of a level set function.
2. The method for establishing a microstructure proxy model as described in claim 1, characterized in that: Virtual microstructures are defined based on the properties of level set functions. : The microstructures corresponding to multiple level set functions are obtained by merging them using a union Boolean operation: After merging, multiple microstructures within the reference domain are combined into a single final microstructure: The resulting periodic microstructure The macroscopic equivalent elasticity tensor is obtained by calculating the homogenization theory: It can be represented in matrix form: The volume fraction of microstructure is obtained from the structural relationship: in For the local elastic tensor of the microstructure, Given the known macroscopic strain field in the horizontal direction. Given the known local strain field in the horizontal direction, The known macroscopic strain field in the vertical direction. Given the known local strain field in the vertical direction. To obtain the elastic tensor matrix representation of the microstructure with a cutting height of H, These are the 9 elements in the elastic tensor. It is the volume fraction. Where is the size of the microstructure, and D represents the size of the entire reference domain.
3. The method for establishing a microstructure proxy model as described in claim 2, characterized in that: Select a series of equidistant cutting heights The equivalent elasticity matrix of the corresponding N sets of reference microstructures is obtained: The initial surrogate model for radial basis functions is represented as follows: in: For radial basis functions, For input sample points, The weight coefficients corresponding to the Nth sample point; Represents sample points eigenvectors, writing: 。 4. The method for establishing a microstructure proxy model as described in any one of claims 1-3, characterized in that: The remaining sample points in the sample point database are selected sequentially, and their corresponding feature vectors are subjected to Schmitt orthogonalization with the feature vectors of the selected final proxy model sample points: in This indicates the number of sample points that have been selected. For the first j The orthogonal eigenvectors of the selected sample points are used; the error introduced by the next sample point is calculated. choose Largest sample point i The next sample point as the final proxy model: Calculate the total error of the selected sample points: The criteria for terminating the selection of sample points are as follows: judge If the requirements are met, the selection of sample points ends; otherwise, S3 is repeated to select the (k+2)th sample point.
5. A structural optimization method, characterized in that: The optimization method uses the microstructure proxy model establishment method described in any one of claims 1-4 to construct a proxy model of the structure to be optimized, and then uses the proxy model to perform structural optimization.
6. A system for establishing a microstructure proxy model, characterized in that: The system includes a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, it performs the method for establishing a microstructure proxy model as described in any one of claims 1-4.
7. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores machine-executable instructions, which, when invoked and executed by a processor, cause the processor to implement the method for establishing a microstructure proxy model as described in any one of claims 1-4 or the structural optimization method as described in claim 5.