Metamaterial structure parameter optimization method based on continuous-binary discrete mixed variables

Abstract

The invention discloses a metamaterial structure parameter optimization method based on continuous-binary discrete mixed variables. The method comprises the following steps: firstly, establishing a metamaterial structure parameterized model which comprises an input variable, a boundary condition and a target function; establishing a kernel function of a continuous-binary discrete mixed variable, wherein the kernel function is used for identifying continuous and binary discrete variables in a Gaussian process; then, an input space is initialized based on Sobol static sampling; constructing a multi-task Gaussian process agent model, and training the agent model; and finally, constructing an acquisition function, and calculating the overall hypervolume lifting amount of the smooth expectation by using the acquisition function to guide the next sampling decision. By monitoring the average hypervolume increase amount, whether optimization needs to increase the number of input vectors or whether the input vectors are converged or not is judged. By constructing an optimization framework suitable for continuous-binary discrete mixed variables, dual-performance optimization under a small sample condition is realized, the optimal structure configuration is efficiently obtained, and the optimization precision and the convergence speed are improved.

Description

Technical Field

[0001] This invention belongs to the field of metamaterial structure optimization design technology, specifically involving a metamaterial structure parameter optimization method based on continuous-binary discrete mixed variables. Background Technology

[0002] Metamaterial structures are typically composed of multiple periodic or quasi-periodic cells. As the basic functional unit that determines macroscopic equivalent properties, the cell exhibits a high degree of configuration diversity. From a structural composition perspective, cells can be composed of a single type of basic mechanical structure, such as negative Poisson's ratio cells, negative stiffness cells, or other typical cells; or they can be composed of multiple substructures with different functions, achieving more complex mechanical responses or multifunctional coupling characteristics through the synergistic effect between substructures. Composite cells provide a natural platform for introducing additional degrees of freedom for control and functional expansion into metamaterial structures.

[0003] Metamaterial structural parameter optimization typically uses structural parameters as input variables and performance as the optimization objective to reconstruct the metamaterial structure. Structural parameters refer to continuous variables with boundary constraints, such as length, angle, thickness, spacing, and aperture. However, for high adaptability to complex excitation and constraint conditions, additional controllability has become an important requirement for the practical application of metamaterial structures. Various methods exist to achieve controllability, but the most direct and common is state switching. This involves switching certain cells or local components (such as cosine beams, origami structures, and other bistable structures) between two mutually exclusive states (such as steady-state switching, mirror / reverse configurations, connection / disconnection, etc.). The state parameters, as binary discrete variables, can participate in the design together with continuous variables, forming a continuous-binary discrete hybrid variable, which serves as the input to the optimization model. However, this leads to a huge design space and increased search difficulty, and traditional optimization strategies struggle to obtain effective solution sets with limited samples.

[0004] Existing metamaterial structural parameter optimization methods primarily target continuous variables. For example, patent application number 202211628539.2 (filed on December 17, 2022) discloses a microstructure optimization design method for a three-dimensional truss structure lattice material. This method addresses more complex mechanical performance optimization problems such as energy absorption rate and structural stability by applying Bayesian optimization and a neural network surrogate model to optimize structural parameters. Patent application number 202311699465.6 (filed on December 11, 2023) discloses a multifunctional coupled metamaterial and its design method for controlling temperature and displacement fields. This method applies Bayesian optimization to achieve directional topology optimization of the metastructure by controlling the coupled temperature and displacement fields. These existing methods do not address state parameters as discrete variables. Therefore, this application proposes a metamaterial structural parameter optimization method based on a continuous-binary discrete mixed variable approach. This method can handle continuous-binary discrete mixed variables, is suitable for dual-performance optimization, and possesses fast convergence capability with small samples, enabling rapid configuration and high-performance design of metamaterial structures under complex conditions. Summary of the Invention

[0005] To address the shortcomings of existing technologies, the technical problem this invention aims to solve is to provide a method for optimizing metamaterial structural parameters based on continuous-binary discrete mixed variables.

[0006] The present invention solves the aforementioned technical problem by adopting the following technical solution: A method for optimizing metamaterial structural parameters based on continuous-binary discrete mixed variables includes the following steps: Step 1: Establish a parametric model of the metamaterial structure, including input variables, boundary conditions, and objective function; The input variables include the main structural parameters and state switching parameters of the metamaterial structure, which together form the input vector. ; (1) in, Indicates the first Line number The first cell of the column A continuous variable, Indicates the first Line number The first cell of the column A binary discrete variable, , This indicates the number of continuous variables and binary discrete variables. , Indicates the number of rows and columns of a cell; The boundary conditions for the input variables are: (2) in, , The states represent the lower and upper limits of the constraint range of continuous variables, and -1 and 1 represent the first and second states of the state-switchable component, respectively. The objective function is: (3) in, and These are two objective functions. and These are, respectively, the objective functions related to performance extracted after mathematical definition or simulation processing; Step 2: Establish a kernel function for the continuous-binary discrete mixed variable, which is used to identify continuous and binary discrete variables in the Gaussian process; The kernel function for a continuous-bivariate discrete mixed variable is: (6) (5) in, The kernel function represents a continuous-bivariate discrete variable. , This represents the values ​​of a continuous-bivariate discrete variable in two evaluations. Kernel functions representing continuous variables, , This indicates the values ​​of a continuous variable in two evaluations. Indicates the weighting coefficient. A kernel function representing a binary discrete variable. , This represents the values ​​of a binary discrete variable in two evaluations. , Indicates the first The values ​​of two binary discrete variables in the two evaluations. Indicates the first Automatic correlation determination of length scale for two discrete binary variables; Step 3: Initialize the input space based on Sobol static sampling; Step 4: Construct a multi-task Gaussian process surrogate model and use the surrogate model to probabilistically model the objective function; select a finite number of input vectors and their corresponding true values ​​of the objective function from the input space to form an initial dataset, and use the initial dataset to train the multi-task Gaussian process surrogate model; During the construction of the surrogate model, a task identifier variable is added to the end of the input vector. The task identifier variable serves as the task index input for the multi-task Gaussian process surrogate model. The value of the task identifier variable is 0 or 1, representing the first and second optimization objectives, respectively. Step 5: Construct the acquisition function and use it to calculate the smoothed expected overall overvolume increase, which will guide the next sampling decision. The acquisition function is expressed as: (9) in, Indicates that the two optimization objectives are in The smoothed expected value of the overall supervolume increase over a given prior random function value This represents the number of posterior random function values. , This represents a smooth intermediate quantity related to the first and second optimization objectives. , Indicates the first The corresponding input vector of the th th The values ​​of the first and second optimization objectives in a plurality of posterior random function values. Indicates an adjustable temperature parameter; Step 6: By monitoring the hypervolume, determine whether the optimization needs to increase the number of input vectors or has already converged; When the end If the average hypervolume improvement in each iteration is lower than the preset threshold, it is considered that the current sampling strategy is stagnating in improving the Pareto front, and the number of input vectors is increased. If the average hypervolume improvement is still lower than the preset threshold after increasing the number of input vectors multiple times, it is determined that the optimization process is converging, and the dominant solution, non-dominated solution, and the optimal input variable values ​​corresponding to the non-dominated solution are output. It is a constant greater than zero.

[0007] Furthermore, the smoothing intermediate quantities related to the optimization objective are calculated using the following formula: (8) in, Indicates the relationship with the first Smoothed intermediate quantities related to the optimization objective Represents a soft addition function. Representing the The current optimal value of each optimization objective. Indicates the first The corresponding input vector of the th th The nth posterior random function value The value of each optimization objective.

[0008] Compared with the prior art, the present invention has the following beneficial effects: This invention targets metamaterial structures composed of state-switching cell units (containing state-switching components). The design space is a mixture of variables, including continuous variables (such as length, angle, thickness, spacing, and aperture) and binary discrete variables (such as steady-state switching, mirror / reverse configurations, connection / disconnection, etc.). By constructing an optimization framework suitable for continuous-binary discrete mixed variables (including parameter identification, surrogate models, and sampling strategies), arbitrary dual-performance optimization (determined according to specific operating conditions) under small sample conditions is achieved. This not only efficiently obtains the optimal structural configuration that satisfies operating constraints but also effectively reduces the computational cost of optimizing complex metamaterial structure parameters, achieving high-precision optimization and rapid convergence. Attached Figure Description

[0009] Figure 1 This is a schematic diagram of the metamaterial structure according to an embodiment of the present invention; Figure 2 This is a schematic diagram of the cell structure according to an embodiment of the present invention; Figure 3 This is an overall flowchart of the present invention; Figure 4 This is a supervolume convergence curve diagram of an embodiment of the present invention; Figure 5 This is a schematic diagram of the Pareto solution according to an embodiment of the present invention; Explanation of reference numerals in the attached drawings: 1. Switchable state component; 2. Butterfly-like expansion structure; 3. Local resonance component. Detailed Implementation

[0010] Specific embodiments are given below with reference to the accompanying drawings. These specific embodiments are only used to further illustrate the technical solution of the present invention and do not limit the scope of protection of this application.

[0011] See Figure 1-2 The metamaterial structure in this embodiment includes Each cell contains two state-switchable components 1, a butterfly-like expansion structure 2, and a local resonance component 3. The local resonance component 3 is embedded in the center of the butterfly-like expansion structure 2. The two state-switchable components 1 are distributed in the upper and lower hollow areas of the butterfly-like expansion structure 2 and are fixedly connected to the butterfly-like expansion structure 2.

[0012] See Figure 3 This invention provides a method for optimizing metamaterial structure parameters based on continuous-binary discrete mixed variables. Each cell of the metamaterial structure includes at least one state-switchable component, and includes the following steps: Step 1: Establish a parametric model of the metamaterial structure, including input variables, boundary conditions, and objective function; The input variables are the main structural parameters and state switching parameters that affect the performance of metamaterial structures; these parameters form the input vector. (1) in, It is an input vector containing all continuous variables and binary discrete variables in all cells. Indicates the first Line number The first cell of the column A continuous variable, Indicates the first Line number The first cell of the column A binary discrete variable, , This indicates the number of continuous variables and binary discrete variables. , The number of rows and columns represents the number of cells; continuous variables refer to structural parameters that can change continuously (such as length, angle, thickness, spacing, aperture, etc.), while binary discrete variables refer to the state parameters of components in the cell that can switch states.

[0013] The boundary conditions for the input variables are: (2) in, , The lower and upper limits of the constraint range of continuous variables are represented, and the binary discrete variables represent the state of the state-switchable component. -1 and 1 represent the first state and the second state (e.g., steady state). The objective function is: (3) in, and These are two objective functions, which take the performance of the metamaterial structure as the optimization objective and are determined according to the actual working conditions, such as stiffness, energy absorption, vibration reduction and other properties. and These are objective functions related to performance extracted through mathematical definition or simulation processing. This embodiment takes vibration reduction performance optimization as an example. and These refer to the lower boundary of the vibration attenuation frequency band extracted after simulation processing and the vibration transmission rate within a specified frequency range, respectively.

[0014] Step 2: Establish a kernel function for the continuous-binary discrete mixed variable, which is used to identify continuous and binary discrete variables in the Gaussian process; To enable Gaussian processes to identify continuous and binary discrete input variables, a kernel function for mixed variables needs to be established. First, the binary discrete variables need to be addressed. Since the discrete input space is non-metric and discontinuous, lacking infinitesimal perturbations and continuity, it cannot be directly identified by Gaussian processes. Therefore, to satisfy the continuity assumption of the input domain in Gaussian processes, whether binary discrete variables take the same value is defined as an effective and interpretable similarity measure. This measure can serve as a positive definite kernel within the Gaussian process framework.

[0015] Bayesian optimization is an iterative search and evaluation of parameters in the parameter space using a Gaussian process. If any two binary discrete variables have the same value, the parameters are considered similar in the two evaluations; if they have different values, they are considered dissimilar. Since the binary discrete variables are independent, the overall similarity can be expressed as the product of the similarities of the individual binary discrete variables. This is equivalent to counting the number of binary discrete variables with different values, i.e., the Hamming distance. The Hamming distance of binary discrete variables is expressed as: (4) in, This represents the two sets of binary discrete variables that need to be evaluated in the Gaussian process. and Hamming distance, , Indicates the first The values ​​of two binary discrete variables in the two evaluations; Equation (4) defines the Hamming distance, which equals 0 when the values ​​of the two discrete variables are the same and equals 1 when the values ​​are different. However, the Hamming distance is still discrete and non-differentiable, and therefore not suitable for direct use in Gaussian processes. To solve this problem, an exponential transformation is applied to construct a smooth, differentiable discrete kernel consistent with the exponential form of the continuous kernel. The kernel function for the two discrete variables is then expressed as: (5) in, A kernel function representing a binary discrete variable. Indicates the first Automatic correlation determination (ARD) length scale for binary discrete variables. ARD captures feature scales across different dimensions to achieve automatic feature selection. This length scale controls the rate at which the correlation in the kernel function decays with increasing input distance.

[0016] For kernel functions of continuous variables, the Matérn kernel function is chosen for controllable smoothness.

[0017] After establishing continuous and discrete kernel functions, a kernel function for a continuous-binary discrete mixture is constructed by linearly weighting the kernel functions for continuous and binary discrete variables. Compared to the multiplicative combination form, the linearly weighted form offers stronger interpretability, more stable training behavior, and better gradient optimization compatibility, while effectively integrating continuous and binary discrete variables. Finally, the kernel function for the continuous-binary discrete mixture is defined as follows: (6) in, The kernel function represents a continuous-bivariate discrete variable. , This represents the values ​​of a continuous-bivariate discrete variable in two evaluations. Kernel functions representing continuous variables, , This represents the values ​​of a continuous variable in two evaluations, with weighting coefficients. Used to balance the relative contributions of the two.

[0018] because It appears explicitly in the kernel function of continuous-bivariate discrete mixtures and directly affects the covariance matrix. ,therefore, As a differentiable hyperparameter, it participates in the marginal log-likelihood (MLL) optimization. Other hyperparameters (such as length scale and output scale) are also optimized through the MLL process. With ARD enabled, each input variable has an independent length scale, thereby enhancing the accuracy of automatic feature selection and prediction. In addition, a small jitter term is added to the diagonal of the covariance matrix. This ensures numerical stability.

[0019] Step 3: Initialize the input space based on the Sobol static sampling method; An initial sampling method based on Sobol quasi-random sequences is adopted to generate several input vectors consisting of continuous variables and binary discrete variables to initialize the input space. Sobol quasi-random sequences have low differential distribution characteristics, which can provide a more stable foundation for the construction of surrogate models and subsequent Monte Carlo (MC) based hypervolume (HV) evaluation.

[0020] Step 4: Construct a multi-task Gaussian process surrogate model, use this model to probabilistically model the objective function, and train the surrogate model; In Bayesian optimization, surrogate models are used to approximate objective functions that are computationally expensive or analytically infeasible, thus achieving accurate predictions without repeatedly evaluating the true objective function. Therefore, a surrogate model for the Gaussian regression process is needed. To address the dual-performance optimization requirement, a multi-task Gaussian process surrogate model (MultiTaskGP) under the BoTorch framework is designed.

[0021] In the surrogate modeling phase, a one-dimensional task identifier variable is introduced at the end of each input vector in the input space to identify and model different optimization objectives. In this embodiment, there are a total of 27 input variables for the metamaterial structure. A task identifier variable is added after each input variable, taking a value of 0 or 1, representing the first and second optimization objectives respectively. Therefore, the input vector is expanded to 28 dimensions. The task identifier variable serves as a task list and as the task index input in the multi-task Gaussian process surrogate model, enabling the model to jointly model different objectives while sharing the input feature space, thus establishing a Gaussian regression process surrogate model with dual performance optimization.

[0022] Based on the aforementioned multi-task Gaussian process surrogate model, given that the objective function can only be obtained through a finite number of real evaluations during optimization, and that the objective function remains unknown to the optimizer in the early iterations after the surrogate model is trained on initial samples, we treat each optimization objective function as a prior random function without any observed data (i.e., before obtaining the true values ​​of the objective function based on the input parameter values). We then apply a Gaussian process prior to the distribution of each optimization objective function in the input space, thus depicting the initial uncertainty of the objective function in the input space and completing the initialization of the multi-task Gaussian process surrogate model. The Gaussian process prior is defined by the mean function and the covariance kernel function, and is called the prior distribution. After introducing observed data, it is updated to the posterior distribution through Bayesian inference.

[0023] Before optimization begins, an initial dataset is formed by designing experiments or random sampling to select a finite number of input vectors and their corresponding true values ​​of the objective function from the input space. The multi-task Gaussian process surrogate model is then trained using the initial dataset.

[0024] Step 5: Construct the acquisition function and use it to calculate the smoothed expected overall overvolume increase, which will guide the next sampling decision. After training the multi-task Gaussian process surrogate model, the surrogate model is updated from a prior random function to a posterior random function. This posterior random function, based on the observed data (i.e., the true values ​​of the objective function corresponding to some input vectors) and its input vectors in the current iteration, characterizes the predicted mean and uncertainty of the two optimization objectives in the input space, providing a complete probabilistic description for the acquisition function.

[0025] Because the multi-task Gaussian process surrogate model introduces additional task identifier variables into the input vector, it causes a dimensionality inconsistency between the sampling phase and the iterative optimization phase of Bayesian optimization. Therefore, the existing Expected Hypervolume Improvement (EHVI) sampling function cannot be used directly. To ensure computational consistency, a scale tracking and automatic correction module is introduced to dynamically insert or delete task identifiers according to the optimizer's requirements, ensuring the smoothness of the optimization process. Furthermore, adjustable temperature parameters are used... An adaptive smoothing mechanism is introduced to dynamically update parameters during optimization, thereby adjusting the exploration-exploitation tradeoff in real time and further enhancing flexibility. The improved acquisition function uses a differentiable approximation based on soft addition to construct the smoothed expected overall hypervolume increase, ensuring its stable application in gradient optimization even with discrete and mixed variables. An intermediate quantity is introduced for each optimization objective to construct the smoothed expected overall hypervolume increase, which is related to the... The intermediate quantities related to the optimization objective are denoted as . It can be described as: (7) in, Representing the The current optimal value of each optimization objective, i.e., the optimal solution generated in the current iteration; Indicates the first The corresponding input vector of the th th The nth posterior random function value The value of the optimization objective is the predicted output of the surrogate model for the objective function under a certain input vector under the posterior distribution, including the prediction mean and uncertainty (variance). To provide continuously differentiable gradient information for all posterior random function values, the smooth intermediate quantity related to the optimization objective, calculated using soft addition, can be expressed as: (8) in, Indicates the relationship with the first Smoothed intermediate quantities related to the optimization objective This represents a soft addition function that is differentiable everywhere and has a continuous derivative. This indicates an adjustable temperature parameter used to control the smoothness of the approximation. Based on Monte Carlo estimation of the posterior random function, the acquisition function in equation (9) is obtained, which characterizes the two optimization objectives in The smoothed expected overall hypervolume increase over each prior random function value (i.e., the predicted output of the surrogate model of the objective function under a certain input vector under the prior distribution) ; (9) in, This represents the number of posterior random function values. , This represents a smooth intermediate quantity related to the first and second optimization objectives. , Indicates the first The corresponding input vector of the th th The values ​​of the first and second optimization objectives in the posterior random function values.

[0026] Equation (8) ensures that the acquisition function remains fully differentiable and is compatible with gradient-based optimization processes.

[0027] After the acquisition function is established, the smooth expected overall hypervolume improvement amount corresponding to the acquisition function is only used to evaluate the potential improvement value of the input vector and determine the next sampling position before sampling. After the true evaluation is completed and the Pareto solution (non-dominated solution) under the current iteration is updated, the optimization process no longer depends on the smooth expected overall hypervolume improvement amount. Instead, it adopts the quantification of the sampling effect based on the hypervolume and its improvement amount, and introduces the subsequent monitoring and convergence judgment mechanism accordingly.

[0028] Step 6: Introduce a monitoring mechanism based on the hypervolume to determine whether the optimization needs to increase the number of input vectors or has already converged; To further improve sampling efficiency and avoid unnecessary iterations, a monitoring mechanism is introduced to dynamically adjust the sampling process driven by the acquisition function and to determine whether the optimization process has converged. If the average hypervolume improvement in each iteration is lower than a preset threshold, the current sampling strategy is considered to have stalled in improving the Pareto front. In this case, the number of input vectors in the input space is increased to improve the estimation accuracy of the Pareto front. If, after multiple increases in the number of input vectors, the average hypervolume improvement remains lower than the preset threshold, the optimization process is considered to be converging. The dominant solution, non-dominated solution, and the optimal input variable values ​​corresponding to the non-dominated solution (Pareto solution) are then output. This strategy achieves balanced exploration in the early stages and adaptively adjusts the sampling density and search size based on convergence in the later stages, maintaining a balance between exploration and convergence in the optimization process.

[0029] See Figure 4 , 5 The hypervolume convergence curve flattens out, indicating successful optimization convergence. The scatter plots of dominated and non-dominated solutions illustrate the Pareto solution. Based on the input variables corresponding to the optimal solution, the metamaterial structure is reconstructed to meet practical operating requirements.

[0030] Any aspects not covered in this invention are applicable to existing technologies.

Claims

1. A method for optimizing metamaterial structure parameters based on continuous-binary discrete hybrid variables, wherein the metamaterial structure involved is composed of multiple cell arrays, and each cell includes at least one state-switchable component; characterized in that, Includes the following steps: Step 1: Establish a parametric model of the metamaterial structure, including input variables, boundary conditions, and objective function; The input variables include the main structural parameters and state switching parameters of the metamaterial structure, which together form the input vector. ; (1) in, Indicates the first Line number The first cell of the column A continuous variable, Indicates the first Line number The first cell of the column A binary discrete variable, , This indicates the number of continuous variables and binary discrete variables. , Indicates the number of rows and columns of a cell; The boundary conditions for the input variables are: (2) in, , The states represent the lower and upper limits of the constraint range of continuous variables, and -1 and 1 represent the first and second states of the state-switchable component, respectively. The objective function is: (3) in, and These are two objective functions. and These are, respectively, the objective functions related to performance extracted after mathematical definition or simulation processing; Step 2: Establish a kernel function for the continuous-binary discrete mixed variable, which is used to identify continuous and binary discrete variables in the Gaussian process; The kernel function for a continuous-bivariate discrete mixed variable is: (6) (5) in, The kernel function represents a continuous-bivariate discrete variable. , This represents the values ​​of a continuous-bivariate discrete variable in two evaluations. Kernel functions representing continuous variables, , This indicates the values ​​of a continuous variable in two evaluations. Indicates the weighting coefficient. A kernel function representing a binary discrete variable. , This represents the values ​​of a binary discrete variable in two evaluations. , Indicates the first The values ​​of two binary discrete variables in the two evaluations. Indicates the first Automatic correlation determination of length scale for two discrete binary variables; Step 3: Initialize the input space based on Sobol static sampling; Step 4: Construct a multi-task Gaussian process surrogate model and use the surrogate model to probabilistically model the objective function; select a finite number of input vectors and their corresponding true values ​​of the objective function from the input space to form an initial dataset, and use the initial dataset to train the multi-task Gaussian process surrogate model; During the construction of the surrogate model, a task identifier variable is added to the end of the input vector. The task identifier variable serves as the task index input for the multi-task Gaussian process surrogate model. The value of the task identifier variable is 0 or 1, representing the first and second optimization objectives, respectively. Step 5: Construct the acquisition function and use it to calculate the smoothed expected overall overvolume increase, which will guide the next sampling decision. The acquisition function is expressed as: (9) in, Indicates that the two optimization objectives are in The smoothed expected value of the overall supervolume increase over a given prior random function value This represents the number of posterior random function values. , This represents a smooth intermediate quantity related to the first and second optimization objectives. , Indicates the first The corresponding input vector of the th th The values ​​of the first and second optimization objectives in a plurality of posterior random function values. Indicates an adjustable temperature parameter; Step 6: By monitoring the hypervolume, determine whether the optimization needs to increase the number of input vectors or has already converged; When the end If the average hypervolume improvement in each iteration is lower than the preset threshold, it is considered that the current sampling strategy is stagnating in improving the Pareto front, and the number of input vectors is increased. If the average hypervolume improvement is still lower than the preset threshold after increasing the number of input vectors multiple times, it is determined that the optimization process is converging, and the dominant solution, non-dominated solution, and the optimal input variable values ​​corresponding to the non-dominated solution are output. It is a constant greater than zero.

2. The method for optimizing metamaterial structural parameters based on continuous-binary discrete mixed variables according to claim 1, characterized in that, The smoothing intermediates related to the optimization objective are calculated using the following formula: (8) in, Indicates the relationship with the first Smoothed intermediate quantities related to the optimization objective Represents a soft addition function. Representing the The current optimal value of each optimization objective. Indicates the first The corresponding input vector of the th th Among the nth posterior random function values, the nth... The value of each optimization objective.

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