Flow field uncertainty correlation mapping and robust optimization method and system for high-consistency aerosol jet printing

Abstract

The present application belongs to the field of intelligent additive manufacturing quality control and fluid mechanics, and provides a flow field uncertainty correlation mapping and robust optimization method and system for high-consistency aerosol jet printing: based on the original geometric structure of the aerosol jet printing head, an aerosol printing flow field simulation model is established, and then combined with the uncertainty parameters, the flow field uncertainty space mapping data set is obtained; a flow field reasoning agent model is constructed by combining a Gaussian process regression model; a multi-objective optimization function based on velocity-pressure field stability is constructed, and the flow field reasoning agent model is used for iterative solution, and finally the optimal process parameters of aerosol jet printing are obtained through the Pareto optimal boundary identification. The present application is based on the organic combination of computational fluid dynamics simulation and uncertainty modeling based on polynomial chaos expansion, and multi-objective global robust optimization algorithm based on Pareto equilibrium, and overcomes the multi-source airflow interference from the bottom aerodynamic mechanism and global space optimization angle.

Description

Technical Field

[0001] This invention belongs to the interdisciplinary field of intelligent additive manufacturing quality control and fluid mechanics, and in particular relates to a flow field uncertainty correlation mapping and robust optimization method and system for high-consistency aerosol printing. Background Technology

[0002] Aerosol jet printing (AJP) plays a crucial role in the fabrication of microelectronics, flexible electronics, and functionally graded materials. As a high-precision additive manufacturing technology, AJP's droplet formation relies heavily on aerodynamic focusing mechanisms. By precisely controlling the injection of carrier gas and sheath gas, the system can form a highly collimated annular focused gas stream inside the nozzle, achieving the deposition of complex electronic component structures with micron-level resolution. This technology demonstrates unique advantages in the fabrication of micro-sensors, embedded circuits, and biomedical devices.

[0003] Despite its potential, the industrial scalability of AJP devices remains limited by deposition consistency and process reproducibility. The morphological integrity of the deposited pattern depends primarily on the focusing stability of the coaxial multiphase jet within the printhead. However, in actual operation, inherent physical fluctuations in the mass flow controller and gas path system cause the input velocities of the carrier gas and sheath gas to easily deviate randomly from their nominal values. These multi-source gas flow disturbances couple and interfere within the complex contraction geometry of the printhead, resulting in severely uneven velocity distribution in the internal flow field and abnormal fluctuations in the pressure gradient. This inherent instability of the purely aerodynamic field directly disrupts the core focusing gas jet morphology, intensifying shear layer turbulence and blurring gas flow boundaries, which in turn manifests as significant defects, particularly dimensional variations and edge overspray. These defects affect the electrical performance of microelectronics and manufacturing yield. Therefore, characterizing, quantifying, and optimizing these random uncertainties are prerequisites for achieving consistent and reliable manufacturing of high-performance AJP devices.

[0004] For the optimization of aerosol printing flow field processes, traditional methods often rely on manual experience or orthogonal experiments to test airflow parameters, which is not only time-consuming and material-intensive, but also difficult to cover high-dimensional process spaces. On the other hand, traditional computational fluid dynamics numerical simulations can only reveal the aerodynamic focusing behavior under ideal nominal conditions, which has significant limitations: deterministic simulations cannot characterize and quantify the spatial evolution of uncertainties driven by random airflow fluctuations. If the traditional Monte Carlo sampling method is used to evaluate such random disturbances, its extremely low convergence speed will lead to an exponential increase in computational costs, which cannot meet the industry's demand for high-throughput process iteration and real-time prediction, resulting in a serious disconnect between pure theoretical simulation and actual dynamic production. However, existing data-driven and process optimization technologies have two significant blind spots when dealing with the above-mentioned complex behaviors: (1) In terms of flow field uncertainty reasoning, traditional surrogate models can only establish a simple mapping from process input to a single scalar index, which is difficult to handle the multidimensional heterogeneous characteristics under the strong coupling of "process parameter fluctuations" and "spatial coordinate changes". Actual flow field disturbances have significant anisotropy and spatial decay characteristics. Traditional models cannot achieve continuous and high-precision prediction of the uncertainty of the entire spatial flow field, resulting in a lack of reliable "optimization guidance" for downstream optimization. (2) In terms of robust process optimization, traditional optimization strategies often adopt single-objective or simple parameter weighting, which seriously ignores the inherent game relationship in the physical field of the gaseous field. That is, under a specific airflow ratio, "velocity field stability (determines the consistency of deposition size)" and "pressure field stability (determines the continuity of ejection)" often restrict each other. Traditional algorithms cannot identify and remove this physical conflict, and are very prone to getting trapped in local optima. As a result, the calculated process parameters are extremely fragile in the actual workshop environment, causing a large fluctuation in the manufacturing qualification rate.

[0005] In summary, overcoming the bottleneck of process robustness caused by random airflow disturbances is crucial for achieving industrial-scale production of aerosol printing. It is necessary to combine computational fluid dynamics and simulation, uncertainty quantification, data-driven modeling, and multi-objective global optimization intelligent algorithms to accurately capture the flow field response characteristics under multi-dimensional coupling of process and space. Furthermore, in complex aerodynamic conflicts, the optimal physical equilibrium point for global system robustness can be quickly and accurately identified. This can effectively improve the stability and repeatability of aerosol printing, ensuring that the produced functional components meet stringent electronic performance, geometric accuracy, and surface quality requirements, thereby promoting its application in high-precision fields such as aerospace and biomedicine. Summary of the Invention

[0006] To address the limitations of existing technologies, such as the inability of deterministic flow field simulation to handle random fluctuations and the high cost of traditional sampling calculations, this invention proposes a flow field uncertainty correlation mapping and robust optimization method for high-consistency aerosol printing. This method organically combines high-fidelity computational fluid dynamics simulation, non-invasive polynomial chaotic expansion, and a four-dimensional Gaussian process regression surrogate model. Specifically, this invention first efficiently quantifies the spatial propagation mechanism of airflow fluctuations within complex nozzles; then, it constructs a process-space multidimensional coupled anisotropic Gaussian process model to achieve real-time inference of the uncertainty characteristics of the global flow field; finally, it overcomes the limitations of traditional single-objective optimization by constructing an evaluation function based on dual flow field uncertainty equilibrium and accurately identifying the Pareto of the global robust process space using a multi-objective optimization algorithm. The optimized process parameters and their fluctuation tolerances are then transformed into equipment control commands under actual production conditions, achieving highly consistent and reliable process output.

[0007] This invention provides the following technical solution: A flow field uncertainty correlation mapping and robust optimization method for high-consistency aerosol printing includes the following steps: Step S1. Based on the original geometry of the aerosol printing nozzle, establish an aerosol printing flow field simulation model. Then, based on the aerosol printing flow field simulation model and combined with uncertainty parameters, obtain the flow field uncertainty space mapping dataset. Step S2. Based on the flow field uncertainty spatial mapping dataset and the Gaussian process regression model, construct a flow field inference proxy model; Step S3. Construct a multi-objective optimization function based on velocity-pressure field stability, and use the flow field reasoning proxy model to perform iterative solution. Finally, obtain the optimal process parameters for aerosol printing through Pareto optimal boundary identification.

[0008] Preferably, step S1 includes: S1.1. For aerosol inkjet printheads, establish an aerosol printing flow field simulation model and conduct physical simulation analysis; among which, the uncertainty quantification parameters include sheath flow velocity and carrier flow velocity; S1.2. Based on the dimensionality reduction sampling strategy, the uncertain quantization parameters defined by quantization are sampled in the random parameter space; and based on the established flow field simulation model, the steady-state response data of the physical field corresponding to each set of input parameters sampled in the random parameter space are extracted. S1.3. Construct orthogonal basis functions for polynomial chaotic expansion based on aerosol printing flow field; S1.4. Based on the physical field steady-state response data and orthogonal basis functions, extract the flow field uncertainty spatial mapping dataset covering the nozzle inlet, junction cavity, and outlet regions.

[0009] Preferably, step S2 includes: S2.1. Within the process parameter space defined by the carrier gas flow rate and sheath gas flow rate, perform Latin hypercube sampling and couple it with the geometric space coordinates of the CFD module to construct a four-dimensional feature dataset containing process and spatial dimensions. S2.2. Construct a Gaussian process regression model based on the Matern kernel function; S2.3. Based on the four-dimensional feature dataset, perform hyperparameter optimization and training on the Gaussian process regression model; S2.4. Based on the Gaussian process regression model optimized and trained by hyperparameters, establish the flow field inference proxy model.

[0010] Preferably, step S3 includes: S3.1. Construct a multi-objective optimization function based on the stability of the velocity-pressure field; S3.2. The multi-objective optimization function is used as the label data for training the flow field inference proxy model. After the physical process parameters are digitized, the genetic algorithm is used for iterative optimization based on the multi-objective optimization algorithm. S3.3. Based on the Pareto equilibrium point obtained by iterative optimization convergence using a genetic algorithm, the most robust inflection point is obtained by identifying the Pareto optimal boundary. Then, the optimal process parameters for aerosol printing are obtained by mapping the target space to the physical process space.

[0011] This invention also provides a flow field uncertainty correlation mapping and robust optimization system for high-consistency aerosol printing, including: a simulation model module, a surrogate model module and a parameter optimization module; The simulation model module is used to establish an aerosol printing flow field simulation model based on the original geometry of the aerosol printing nozzle, and then, based on the aerosol printing flow field simulation model and combined with uncertainty parameters, obtain a flow field uncertainty space mapping dataset. The proxy model module is used to construct a flow field inference proxy model based on the flow field uncertainty spatial mapping dataset and the Gaussian process regression model; The parameter optimization module is used to construct a multi-objective optimization function based on the stability of the velocity-pressure field, and to perform iterative solution using the flow field inference proxy model. Finally, the optimal process parameters for aerosol printing are obtained through Pareto optimal boundary identification.

[0012] Preferably, the simulation model module includes: a model building unit, a sampling unit, an orthogonal basis function unit, and a dataset unit. The model building unit is used to establish an aerosol printing flow field simulation model for the aerosol printing nozzle and to perform physical simulation analysis; wherein, the uncertainty quantification parameters include sheath flow velocity and carrier flow velocity. The sampling unit is used to sample the quantized uncertain parameters in the random parameter space based on the dimensionality reduction sampling strategy; and to extract the physical field steady-state response data corresponding to each set of input parameters sampled in the random parameter space based on the established flow field simulation model. The orthogonal basis function unit is used to construct orthogonal basis functions based on the polynomial chaotic expansion of the aerosol printing flow field; The dataset unit is used to extract a spatial mapping dataset of flow field uncertainty covering the nozzle inlet, junction cavity, and outlet regions based on the physical field steady-state response data and orthogonal basis functions.

[0013] Preferably, the proxy model module includes: a coupling unit, a Gaussian model unit, a Gaussian training unit, and an encapsulation unit; The coupling unit is used to perform Latin hypercube sampling in the process parameter space defined by the carrier gas flow rate and the sheath gas flow rate, and to couple with the geometric space coordinates of the CFD module to construct a four-dimensional feature dataset containing process dimension and spatial dimension. The Gaussian model unit is used to construct a Gaussian process regression model based on the Matern kernel function; The Gaussian training unit is used to perform hyperparameter optimization and training on the Gaussian process regression model based on the four-dimensional feature dataset. The encapsulation unit is used to establish the flow field inference proxy model based on the Gaussian process regression model optimized and trained by hyperparameters.

[0014] Preferably, the parameter optimization module includes: an objective function unit, an iterative optimization unit, and a Pareto optimization unit; The objective function unit is used to construct a multi-objective optimization function based on velocity-pressure field stability; The iterative optimization unit is used to use the multi-objective optimization function as the label data for training the flow field inference proxy model. After digitizing the physical process parameters, it performs iterative optimization using a genetic algorithm based on the multi-objective optimization algorithm. The Pareto optimization unit is used to find the Pareto equilibrium point obtained by iterative optimization based on the genetic algorithm, identify the most robust inflection point by using the Pareto optimal boundary, and then obtain the optimal process parameters for aerosol printing by mapping the target space to the physical process space.

[0015] The beneficial effects of this invention are as follows: This invention provides a flow field uncertainty correlation mapping and robust optimization method and system for high-consistency aerosol printing, which has the following advantages compared with traditional solutions: (1) The flow field uncertainty model driven by the aerosol printing fluid simulation model breaks through the cognitive limitations and sampling computing power bottleneck of traditional deterministic simulation. It can accurately quantify the nonlinear evolution law of random fluctuations of multi-source airflow in complex nozzle structure, and provide high-fidelity and high-efficiency underlying physical support for subsequent process optimization. (2) Construction of a flow field uncertainty proxy model based on a four-dimensional Gaussian process: In the flow field state reasoning stage, the dependence on time-consuming physical simulation in the optimization iteration is completely eliminated, and the high-speed and continuous prediction of the uncertainty characteristics of the spatiotemporally coupled flow field in the whole domain is realized. (3) Based on the equilibrium of uncertainty in dual flow fields, the robust process optimization of aerosol printing overcomes the shortcomings of conventional single-objective optimization which makes it difficult to balance the physical conflict of aerosols. It accurately extracts the Pareto optimal boundary of the entire process space, thereby locking the optimal physical equilibrium point of the system robustness and fundamentally suppressing the morphological degradation of the core airflow focusing beam.

[0016] The combination of computational fluid dynamics simulation and polynomial chaotic expansion uncertainty modeling, high-speed proxy inference of flow field characteristics based on high-dimensional Gaussian processes, and multi-objective global robust optimization algorithm based on Pareto equilibrium overcomes multi-source airflow interference from the perspective of underlying aerodynamic mechanism and global spatial optimization, greatly improving the high consistency and process robustness of printing complex micro and nano devices. Attached Figure Description

[0017] To more clearly illustrate the technical solution of the present invention, the drawings used in the embodiments are briefly described below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort. Figure 1 This is an overall flowchart of a flow field uncertainty correlation mapping and robust optimization method for high-consistency aerosol printing provided in an embodiment of the present invention; Figure 2 The following is a schematic diagram of the simulation model of an embodiment of the present invention. (ad) respectively represent the key simulation flow field regions selected for uncertainty quantification and optimization analysis, wherein (a) represents the carrier gas inlet region, (b) represents the sheath gas inlet region, (c) represents the combined cavity region, and (d) represents the nozzle outlet region. Figure 3 In this embodiment of the invention, the collected input flow velocity is quantized into a distribution containing Gaussian uncertainty. (a) represents the Gaussian uncertainty contained in the sheath flow velocity, and (b) represents the Gaussian uncertainty contained in the carrier flow velocity. Figure 4This is a schematic diagram of the experimental design points for the sparse grid method in an embodiment of the present invention. (ac) represent the design levels of grid points as 1, 2, and 3, respectively, and the number of grid points increases with the level. Figure 5 This is a schematic diagram of the orthogonal polynomial expansion used in the embodiments of the present invention. (a) is a 1D Hermite orthogonal polynomial, and (bd) is a 2D Hermite orthogonal polynomial. Figure 6 This is a schematic diagram of the stochastic characterization of the gas inlet flow field when the input uncertainty is 5% in an embodiment of the present invention. (a1, b1) represent the average value and standard deviation of the velocity field at the sheath gas inlet, respectively; (c1, d1) represent the average value and standard deviation of the pressure field at the sheath gas inlet, respectively; (a2, b2) represent the average value and standard deviation of the velocity field at the carrier gas inlet, respectively; and (c2, d2) represent the average value and standard deviation of the pressure field at the carrier gas inlet, respectively. Figure 7 This is a schematic diagram of the stochastic characterization of the cavity flow field when the input uncertainty is 5% in an embodiment of the present invention. (ab) represent the mean and standard deviation of the velocity field, respectively, and (cd) represent the mean and standard deviation of the pressure field, respectively. Figure 8 This is a schematic diagram of the stochastic characterization of the nozzle outlet flow field when the input uncertainty is 5% in an embodiment of the present invention. (ab) represent the mean and standard deviation of the velocity field, respectively, and (cd) represent the mean and standard deviation of the pressure field, respectively. Figure 9 This is a schematic diagram of residual convergence analysis in an embodiment of the present invention. In the diagram, (a) represents the absolute residual of the model after modeling with secondary grid points and primary grid points, and (b) represents the absolute residual of the model after modeling with tertiary grid points and secondary grid points. The results show the convergence of the model. (a1,b1) represents the average velocity of the airflow in the cavity, (a2,b2) represents the standard deviation of the airflow velocity in the cavity, (a3,b3) represents the average pressure of the airflow in the cavity, and (a4,b4) represents the absolute residual of the standard deviation of the airflow pressure in the cavity. Figure 10 The test points are obtained by sampling design in the working space of the carrier gas flow and the sheath gas flow using the Latin hypercube sampling (LHS) method in the embodiments of the invention. Figure 11 The following diagrams illustrate the performance of training a surrogate model for the uncertainty of the flow field using the Gaussian process regression method: (a) shows the training process of the standard deviation model for the velocity field; (b) shows the training process of the standard deviation model for the pressure field; (c) shows the performance test diagram of the standard deviation model for the velocity field; and (d) shows the performance test diagram of the standard deviation model for the pressure field. Figure 12Based on the developed Gaussian process surrogate model, a random input of a set of working parameters (carrier gas velocity, sheath gas velocity) yields an example diagram of the uncertainty distribution of the corresponding flow field. (a1-d1) represent the standard deviation distribution of the velocity fields at the inlet of the carrier gas field, the inlet of the sheath gas field, the junction cavity, and the outlet, respectively; (a2-d2) represent the standard deviation distribution of the pressure fields at the inlet of the carrier gas field, the inlet of the sheath gas field, the junction cavity, and the outlet, respectively. Figure 13 The schematic diagram for binary encoding of working parameters (sheath gas velocity, carrier gas velocity) helps to combine with Gaussian process regression surrogate model for multi-objective intelligent optimization; Figure 14 A complete workflow diagram for combining a Gaussian process regression surrogate model with a multi-objective optimization algorithm to perform multi-objective optimization (while reducing uncertainty disturbances in the velocity and pressure fields); Figure 15 The diagram shows the results of multi-objective optimization. (a) represents the Pareto front obtained by optimization; (b) represents the set of parameters identified after optimization and a set of optimal parameters corresponding to the point of maximum curvature. Figure 16 The optimal operating parameters represent the uncertainty of the optimized flow field. (a1-d1) represent the standard deviation distribution of the velocity fields at the inlet of the carrier gas field, the inlet of the sheath gas field, the junction cavity, and the outlet, respectively. (a2-d2) represent the standard deviation distribution of the pressure fields at the inlet of the carrier gas field, the inlet of the sheath gas field, the junction cavity, and the outlet, respectively. Figure 17 To verify the comparison of the experimental results, (a1-a5) represents the samples printed using the optimal parameters identified, which have high consistency; (b1-b5) represents the samples printed using the original parameters identified, which have poor consistency and large quality fluctuations. Detailed Implementation

[0018] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0019] Example 1 This invention provides a flow field uncertainty correlation mapping and robust optimization method for high-consistency aerosol printing, the overall scheme of which is as follows: Figure 1 As shown, the main technologies involved are explained as follows: (1) Computational Fluid Dynamics (CFD) Simulation Module: Method Description: Based on the finite volume method, the Navier-Stokes equations (NS equations) are solved to perform a three-dimensional numerical simulation of the complex gas-solid two-phase flow inside the nozzle.

[0020] Function and Purpose: As the physical truth engine of the system, it obtains the original response data of the flow field under different disturbances through high-precision simulation, providing the underlying physical basis for subsequent modeling.

[0021] (2) Smolyak Sparse Grid Sampling Method Description: This is an efficient sampling criterion for high-dimensional spaces. By selecting a specific combination of tensor product grids, it avoids the "curse of dimensionality" that traditional grids suffer from as the dimension increases.

[0022] Function and purpose: In the first stage, it covers the random input space of airflow fluctuations to the maximum extent with very few CFD simulations, greatly improving modeling efficiency.

[0023] (3) Non-invasive polynomial chaos expansion (PCE) Method Description: A method for representing random responses as orthogonal polynomial basis function series with respect to input variables, which can analyze complex stochastic processes.

[0024] Function and purpose: To realize the transformation from discrete simulation points to continuous probability distribution, directly extract the mean and variance of the global flow field analytically, and construct a quantitative model of the uncertainty of the flow field.

[0025] (4) Latin hypercube sampling (LHS) Method Description: An efficient experimental design method with key advantages including efficient coverage of the input space, reduced sample size, avoidance of sample concentration, applicability to high-dimensional problems, and global exploration and optimization of the working parameter space of multiple factors with minimal experimental effort.

[0026] Function and Purpose: Used in the second stage for sample design in the process parameter space to ensure that the training data has extremely high representativeness and coverage throughout the entire process window.

[0027] (5) Regression of anisotropic Gaussian processes Method Description: A nonparametric probabilistic model based on Bayesian inference, whose "anisotropic" feature allows the model to configure independent correlation scales for inputs of different dimensions (such as flow and coordinates).

[0028] Function and purpose: As a high-speed surrogate model, it provides real-time prediction of flow field characteristics while maintaining high physical fidelity.

[0029] (6) Multi-objective evolutionary algorithm NSGA-III Method description: Simulate the crossover, mutation and selection mechanisms in the process of biological evolution to find a balance among multiple conflicting objectives.

[0030] Function and application: The drive system automatically seeks optimization within the entire process space and identifies the Pareto non-dominated solution set that can simultaneously take into account "focusing accuracy" and "internal pressure stability".

[0031] (7) Pareto Balance and Inflection Point Decision Method description: The Pareto front represents the optimal physical limit of the system under multi-objective game; the "KneePoint" is the equilibrium position with the most dramatic curvature change and the highest overall cost-effectiveness.

[0032] Function and Purpose: To mathematically resolve the coupling conflict between the uncertainties of velocity and pressure, lock in the most robust golden process point, and output the final control decision.

[0033] Below, in conjunction with Figure 1 The technical solution of the present invention will be described in detail below: Step S1. Modeling of flow field uncertainties driven by aerosol printing fluid simulation model The main work includes: establishing an aerosol printing flow field simulation model based on the original geometry of the aerosol printing nozzle; and then, based on the aerosol printing flow field simulation model and incorporating uncertainty parameters, obtaining a flow field uncertainty spatial mapping dataset. Specifically, it consists of the following sub-steps: S1.1. Development of Aerosol Printing Flow Field Simulation Model and Definition of Input Uncertainty (1) Physical Modeling and Mesh Generation: The original geometry of the aerosol jet (AJP) printhead consists of an air inlet, a combined cavity, a connecting cavity, and a nozzle. This invention uses a combination of calipers and a digital microscope to measure the external dimensions of the printhead and X-ray computed tomography to measure the internal dimensions, establishing an aerosol printing flow field simulation model. After determining the original simulation model, the preprocessor ICEM software (Ansys) is used. Mesh the model geometry, such as... Figure 2 As shown. To improve computational accuracy, the simulation model was discretized into a grid primarily composed of quadrilaterals. First, the overall grid was moderately smoothed and refined to ensure its basic quality. To accurately capture key flow characteristics, especially boundary layer effects and strong shear flow within the central mixing chamber, local mesh refinement was applied to the gas confluence region inside the nozzle and the nozzle contraction section.

[0034] (2) Setting of physical field governing equations: The gas inside the nozzle is assumed to be a compressible continuous medium that satisfies the ideal gas law. Under steady-state conditions, the evolution of the system flow field strictly follows the following conservation governing equations: Mass conservation equation (continuity equation): (1) Momentum conservation equation (Navier-Stokes equation): (2) Energy conservation equation: (3) Ideal gas law: (4) In the above formula, Let be the gas density (kg / m³), and t be the time (s). Let the gas velocity vector be (m / s). The flow field pressure (Pa) Let be the gas dynamic viscosity (Pa·s), e be the internal specific energy (J / kg), k be the fluid thermal conductivity (W / (mK)), and T be the fluid temperature. Let be the dissipation function. Here, R is the specific heat capacity at constant volume, and R is the ideal gas constant. and These are the momentum and energy source terms, respectively.

[0035] (3) Assumptions of the numerical model: ① A two-dimensional axisymmetric domain is adopted. The effectiveness of the two-dimensional symmetric model is demonstrated by previous observations of the quasi-axisymmetric flow in the print head; ② Both the sheath gas inlet and the carrier gas inlet are set as velocity inlet boundary conditions.

[0036] (4) Input Random Variable Modeling: The aerodynamic focusing mechanism is mainly controlled by the sheath flow rate (SHGFR) and carrier flow rate (CGFR), which are uncertain quantification parameters. However, due to the inherent randomness of gas pipeline pressure and mass flow rate control, these flow rates typically exhibit random fluctuations centered around their nominal values. To explain this variability, they are defined as mutually independent continuous random variables. For example... Figure 3 As shown, in the simulation, the airflow parameters are set to follow a Gaussian distribution, denoted as the input random vector. In this context, SHGFR and CGFR are both measured in sccm (standard cubic centimeters per minute). Furthermore, the impact of input noise intensity was quantified: based on the quantitative analysis, the standard deviation of the nominal flow rate affected by Gaussian noise fluctuations was determined to be... =5%.

[0037] S1.2. Design of Random Parameter Spatial Sampling and Automated Parallel Computing Traditional uncertainty quantification typically employs Monte Carlo random sampling or the Full Tensor Product (FTP) grid method. However, the former has extremely slow convergence speed, requiring tens of thousands of CFD simulations; the latter suffers from an exponential increase in computational cost (i.e., the "curse of dimensionality") as the dimensionality of the input variables increases. To achieve controllable computational cost while maintaining approximation accuracy, this invention designs an automated high-throughput simulation pipeline based on a dimensionality reduction sampling strategy, specifically including the following design steps: (1) Dimensionality reduction sampling strategy and experimental design in random parameter space: In order to characterize the uncertainty of airflow input, this embodiment adopts the sparse grid strategy based on the Smolyak criterion in the two-dimensional random parameter space for experimental design. Figure 4 Specifically, it demonstrates a sparse grid sampling point combination in a two-dimensional random parameter space for a set of highly volatile and uncertain parameters in aerosol printing—the "carrier gas and sheath gas flow rates." The definition and core advantage of the Smolyak sparse grid lie in its construction of a multidimensional quadrature grid through a linear combination of a set of low-order one-dimensional tensor products. It selectively prunes high-order interactive nodes that have a smaller coupling effect on the random response of the flow field, thereby reducing the total number of nodes required for configuration. Keep it at an extremely low level. For example... Figure 4 As shown, this combination of sampling points expanded according to specific orthogonal rules in the random parameter space constitutes the numerical basis for quantifying the propagation of uncertainty. By evaluating deterministic flow field models at these representative nodes, the uncertainty distribution of input parameters (such as airflow fluctuations) can be accurately mapped and transformed into the uncertainty quantification results of output flow field characteristics (such as velocity and pressure fields) with extremely high algebraic accuracy and minimal computational cost (such as extracting the mean and variance of the flow field response).

[0038] (2) Gauss-Hermite Quadrature Node Generation and Physical Mapping: Based on the statistical prior characteristic that the carrier gas flow rate and sheath gas flow rate follow a Gaussian distribution, this invention preferentially uses the Gauss-Hermite quadrature rule when constructing the sparse grid. Under this rule, nodes (i.e., sampling points) are configured in standard normal space. Instead of being randomly generated, these are strictly defined as the roots of the corresponding order Hermitian polynomials. This mathematical property ensures the highest algebraic accuracy when calculating integrals with Gaussian probability density weighting functions. Then, the configuration nodes in the standard normal space are mapped to the actual physical flow space. Let the standard node components be... Then its actual corresponding carrier gas physical flow rate Physical flow rate of sheath gas The mapping relationship is defined as follows: (5) (6) in, and These represent the average nominal flow rates of the carrier gas and sheath gas, respectively. and These represent the standard deviations of the carrier gas and sheath gas flow rate fluctuations, respectively. and Representing the first The standard normal distribution components of the dimensionless configuration nodes in the carrier gas and sheath gas dimensions.

[0039] (3) Construction of an automated high-throughput simulation pipeline: Based on the physical flow parameter combination set generated after mapping, the simulation process is fully automated using the Journal scripting language built into the ANSYS Fluent solver. This pipeline can automatically parse the parameter set, dynamically modify the velocity boundary conditions of the carrier gas and sheath gas inlets in the CFD module, initialize the flow field, and drive the solver (ANSYS Fluent) in a high-performance computing cluster to perform parallel steady-state solutions, thereby improving the simulation efficiency.

[0040] (4) Accurate extraction of steady-state response data of physical fields: for each sampling point in the configured node set Once the CFD calculation meets the convergence criteria for momentum and continuity residuals, the pipeline automatically triggers a data extraction process based on CFD-Post macro commands and the Fluent standard data export interface (Export API). For the extreme value region of interest from the nozzle interior to the nozzle exit, global spatial coordinates are extracted. The results are based on the steady-state response data of the physical field. Specifically, the velocity at the center point of each discrete grid is extracted. and pressure This establishes a one-to-one correspondence between "random input samples" and "high-dimensional flow field spatial response" at the underlying level, providing sufficient deterministic data samples for subsequent polynomial chaotic series expansion (PCE).

[0041] S1.3. Constructing orthogonal basis functions based on polynomial chaotic expansion (PCE) of aerosol printing flow field To quantify the propagation of input variability, the PCE method is used to represent the quantity to be analyzed by projecting onto an orthogonal polynomial, such as... Figure 5 As shown. Specifically, let Let be a vector representing independent standard random variables with uncertain input parameters, in spatial geometric coordinates. At that point, the flow field response output Spectral expansion can be performed based on orthogonal polynomials: (7) in This represents the deterministic spectral coefficients to be calculated. These are multivariate orthogonal polynomial basis functions. Based on the Wiener-Askey scheme, they are used for input random variables that follow a normal distribution. Hermite polynomials are chosen to construct the basis functions of this multivariate orthogonal polynomial. These basis functions satisfy the condition regarding the Gaussian probability density function. Weighted orthogonality condition: (8) in, These represent the first and second parts of the polynomial chaotic expansion. Item and the Multivariate orthogonal polynomial basis functions; Represents standard input random variables The joint probability density function; The symbol for Kronecker.

[0042] To ensure both approximation accuracy and computational efficiency, the total number of truncated terms in the series is... Dimension of random variables With the highest order of the polynomial Strictly defined, the calculation formula is as follows: (9) S1.4. Analysis and Extraction of Statistical Characteristics of Aerosol Printing Flow Field Within a non-invasive spectral projection framework, deterministic spectral coefficients The theoretical solution is given by the Galerkin projection integral: (10) Since high-dimensional integrals are difficult to solve directly, the continuous integral in formula (10) is transformed into a discrete numerical quadrature formula to efficiently calculate the expansion coefficients of each grid point in space: (11) in, Indicates the relationship with the first The product weights corresponding to each Smolyak sparse grid sampling point (configuration node); N This represents the total number of sparse grid sampling points generated. The deterministic physical response term in the pedigree projection integral. This term represents the steady-state solution of the fluid dynamics corresponding to the i-th sample point in the random parameter space, i.e., the global coordinates obtained through deterministic CFD solution. The steady-state field distribution (such as velocity or pressure field) at a location.

[0043] Statistical feature space mapping: Utilizing the mathematical properties of orthogonal polynomials, directly from the solved spectral coefficient set... Analytical analysis is used to obtain the statistical moments of the aerodynamic flow field: The spatial mean (expectation) of the flow field at all coordinate points is analyzed as follows: (12) The spatial variance of the flow field at all coordinate points is analyzed as follows: (13) Extract the mean value under the above global coordinates With variance This results in a "flow field (process-coordinate) uncertainty spatial mapping dataset" covering the nozzle inlet, connecting cavity, and outlet regions. A schematic diagram of the stochastic representation of the gas inlet flow field, connecting cavity flow field, and nozzle outlet flow field when the input uncertainty is 5% is shown below. Figures 6-8 As shown in the figure, this dataset not only accurately reveals the physical fluid mechanism by which minute airflow fluctuations are nonlinearly amplified under complex geometric constraints, but also serves as a high-precision dataset, which is directly output to step S2 for training the high-dimensional Gaussian process surrogate model.

[0044] To verify the approximation accuracy and convergence of the constructed polynomial chaotic expansion (PCE) model when quantifying flow field uncertainties, this invention performs residual convergence analysis on the sparse mesh reconstruction results at different levels, specifically as follows: Figure 9 As shown, a comparison of the two sets of residual contour plots clearly demonstrates that as the sparse grid level increases, the absolute residual amplitudes of various statistical characteristics of the velocity and pressure fields significantly decrease and tend to reach minimum values. This residual convergence analysis fully proves that the "flow field uncertainty spatial mapping dataset" extracted based on the dimensionality reduction sampling strategy of this invention has extremely high numerical convergence and physical fidelity, thus providing a reliable data foundation for the subsequent construction of high-dimensional surrogate models.

[0045] Step S2. Construction of a flow field uncertainty surrogate model based on a four-dimensional Gaussian process (GPR) This phase aims to construct a multi-dimensional correlation mapping system spanning the "process parameter space - geometric-physical space". Based on the flow field uncertainty dataset in the two-dimensional geometric space obtained in step S1, further sampling is performed in the two-dimensional process space to combine the two into a four-dimensional input space. Utilizing the nonparametric probabilistic modeling characteristics of Gaussian process regression (GPR) and its anisotropic kernel function design, a continuously distributed flow field inference proxy model covering the entire process window is constructed. For any combination of input parameters within the process parameter space, this model can characterize the corresponding flow field uncertainty evolution law in real time, thus providing an efficient inference engine for subsequent real-time global optimization.

[0046] S2.1. Process-Space Co-sampling and Four-Dimensional Feature Construction Based on LHS: Process Space Sampling (1) Within the process parameter space defined by carrier gas flow rate and sheath gas flow rate, Latin hypercube sampling (LHS) is performed to ensure that the sampling points have optimal uniformity and representativeness within the multidimensional process window. A schematic diagram of the sampling test points based on LHS is shown below. Figure 10 As shown.

[0047] (2) Four-dimensional feature fusion: The collected process sample points are deeply coupled with the geometric space coordinates (x,y) of CFD, and the flow field statistics (such as the prediction variance of velocity and pressure) extraction method in step S1.4 are associated to construct a four-dimensional feature dataset containing "process dimension" and "space dimension" as input data for training the Gaussian process regression model in S2.2.

[0048] S2.2. Design of an anisotropic Gaussian process model with multidimensional coupling between process and space (1) In response to the inherent physical nature of the inconsistent contribution of process variables (carrier gas and sheath gas flow rates) and geometric spatial coordinates (x, y axis positions) to flow field stability in aerosol printing, this invention abandons the traditional isotropic model and designs an anisotropic Matern kernel function with Automatic Relevance Determination (ARD) functionality. This design allows the Gaussian process regression model to automatically identify and assign different physical weights to each input variable by adjusting the correlation scale of different dimensions during training, thereby significantly improving the model's nonlinear mapping capability under strong sheath gas constraints.

[0049] (2) The anisotropic covariance function constructed in this invention The mathematical expression is defined as follows, used for mathematical model expression and weighted distance definition. (14) in, The signal variance characterizes the overall fluctuation intensity of a random process; d The weighted Mahalanobis distance, a core feature used in this invention, is calculated using the following formula: (15) This formula introduces a scale parameter to transform input features with different dimensions and physical meanings into a unified weighted space for measurement.

[0050] (3) In the above mathematical system, the input vector is defined as X Corresponding process parameters and geometric coordinates These are the anisotropic correlation scale parameters. By configuring independent correlation scales for these four dimensions, the model can accurately capture the "unequal weighting" characteristics of the impact of drastic fluctuations in process parameters (such as sudden changes in sheath gas) and slow changes in spatial location on flow field stability, thereby extracting the most robust feature patterns in complex fluid modeling.

[0051] S2.3. Model Hyperparameter Optimization and Training Based on Full Sample Data Gaussian process regression (GPR) is a nonparametric probabilistic model based on a Bayesian framework. Its core assumption is that the objective function f(x) follows a Gaussian process, meaning that the function values ​​for any set in the input space follow a joint normal distribution. Unlike traditional regression methods, GPR's mathematical essence lies in constructing a covariance matrix among input samples using a kernel function, leveraging the physical correlations of known data points to infer the response in unknown regions. This characteristic makes it exhibit excellent robustness in modeling aerosol flow fields with small samples, high dimensionality, and complex physical fluctuations.

[0052] The specific training steps for the Gaussian process regression model are as follows: (1) Constructing the log-marginal likelihood function. The four-dimensional feature dataset constructed in S2.1 is introduced into the Gaussian process regression model. Based on Bayes' principle, a log-marginal likelihood function is established as the optimization objective. This function is mathematically defined as follows: balancing the data fitting term (Model Fit) and the model complexity term (Complexity Penalty): (16) in, The marginal likelihood probability density function of the target response, This represents the constructed four-dimensional input feature matrix. It includes hyperparameters (such as anisotropic related scales) The covariance matrix of ) The transpose operator for a matrix or vector. yes The response of the established field.

[0053] (2) Maximum Likelihood Estimation and Gradient Optimization of Hyperparameters. The optimal hyperparameters are determined by maximizing the likelihood function described above using the maximum likelihood estimation method. Specifically, the high-performance optimization algorithm of the conjugate gradient method is invoked to optimize each independent scale parameter in the four-dimensional input. Iterative optimization is performed. Through an automatic correlation discovery mechanism, the model dynamically adjusts the weights of each dimension until it finds the parameter combination that best explains the physical fluctuations in the flow field.

[0054] (3) Model generalization performance evaluation and accuracy verification. After training, the mean squared error (MSE) was first used to quantitatively evaluate the training performance of the surrogate model, quantifying the deviation between the model's predicted values ​​and the true values. Based on this, the correlation coefficient R between the predicted results and the simulated true values ​​within the global process window was further calculated to verify whether the model accuracy met the design requirement of R>0.95. Through this threshold verification, it was confirmed that the Gaussian process surrogate model possesses extremely high fidelity, sufficient to support subsequent robust process optimization decisions. The training and testing performance of this model is as follows: Figure 11 As shown.

[0055] S2.4. Development of a Real-Time Inference Model for Uncertainty Characteristics of the Global Flow Field This phase aims to transform the trained high-performance Gaussian process regression (GPR) model into an inference engine with industrial-grade real-time response capabilities, enabling instant mapping from process parameters to the uncertainty distribution of physical fields.

[0056] (1) Lightweight encapsulation of the inference engine. First, the mathematical structure of the model optimized by maximum likelihood estimation in stage S2.3 (including the optimal kernel function, hyperparameters, and weight matrix) is solidified. Redundant iterations in the training stage are eliminated through pre-computation technology, thus obtaining the encapsulated flow field inference proxy model. This proxy model can directly perform matrix-vector operations, bypassing the cumbersome mesh generation and partial differential equation iterative solution process of traditional CFD, significantly improving computational efficiency.

[0057] (2) Cross-space multidimensional parameter mapping and low-latency output. This flow field inference proxy model establishes an efficient input-output mapping mechanism. The system only needs to receive the input carrier gas flow rate and sheath gas flow rate. This allows the call to of a fixed anisotropic kernel function to perform fast matrix mapping on a batch of global geometric coordinate points (x,y), simultaneously generating the predicted variances of the velocity and pressure fields. This mechanism enables high-frequency, real-time coverage of the uncertainty characteristics of the global physical field, from discrete process points.

[0058] (3) Real-time dynamic visualization and robustness assessment feedback. The uncertainty data of the global velocity field and pressure field obtained from the flow field inference proxy model are transformed into intuitive dynamic cloud maps. Through efficient data throughput, this module displays the fluctuation characteristics distribution of the flow field under different process ratios in real time, provides operators with instantaneous "process-quality" early warning, and provides a system evaluation closed loop for the global multi-objective optimization in step S3. Figure 12 The diagram shows the field results obtained based on the flow field reasoning proxy model developed in this stage, after randomly sampling a set of input process parameter combinations.

[0059] Step S3. Optimization of Robust Aerosol Printing Process Based on Dual Flow Field Uncertainty Equilibrium Step S3 aims to transform the aforementioned uncertainty quantification conclusions of the dual flow fields into practical process control criteria. Addressing the stochastic fluctuation characteristics of the coupled flow fields in aerosol printing, this study constructs a complete robust process optimization framework. First, a multi-objective optimization function is defined by integrating velocity field concentration and pressure field stability. Subsequently, a high-precision Gaussian process regression (GPR) flow field inference proxy model constructed in stage S2 is used, combined with a multi-objective optimization algorithm, to perform global optimization of the process parameter space. Based on this, by identifying the Pareto optimal boundary, the competition and equilibrium mechanism between process parameters and printing robustness is revealed, ultimately establishing a set of reliable Pareto equilibrium decision criteria, providing theoretical basis and process guidance for high-precision, high-stability aerosol additive manufacturing.

[0060] S3.1. Construction of a multi-objective optimization function based on velocity-pressure field stability (1) Area-weighted measure of local uncertainty in the region First, regarding the three core regions in the flow field (inlet) , combined cavity ,exit Define the physical quantities (velocity) within each region. U or pressure P The average uncertainty of the results is as follows. To eliminate the influence of non-uniform grid partitioning on the statistical results, the grid area weighted integral method is adopted: For any region Its mean standard deviation Defined as: (17) in, For the first i The standard deviation at each grid point (derived from the analysis results of uncertainty quantification). This represents the area of ​​the grid cell. This metric precisely measures the average intensity of random fluctuations within this physical region.

[0061] (2) Hierarchical integration of total uncertainty in the global velocity field By weighted and integrated velocity field fluctuations in the three regions, a comprehensive index reflecting the transport stability of the entire flow field is constructed. Different regions have different weights in influencing the flow field trajectory. For example, the stability of the velocity field in the exit region directly determines the printing accuracy. Therefore, the weight allocation needs to reflect the physical focus, with a larger weight in the exit region. (18) in, , These represent the area-weighted values ​​of the velocity standard deviation for the inlet region, the junction cavity region, and the outlet region, respectively. , , These represent the weight coefficients corresponding to the three regions mentioned above, and satisfy the following conditions: .

[0062] (3) Hierarchical integration of total uncertainty in the global pressure field Similarly, a comprehensive index is constructed for the pressure stability of the entire flow field. The uncertainty in the pressure field mainly reflects the fluctuations in the flow regime and energy loss. Regional weighting can capture pressure abrupt changes within the binding cavity and pressure rebound fluctuations at the outlet. (19) in, As a core indicator for measuring the macroscopic stability of the flow field. , , These represent the area-weighted standard deviations of pressure in the inlet region, the connecting cavity region, and the outlet region, respectively. , , These represent the weighting coefficients for the three regions mentioned above. The parameter configuration focuses on the junction cavity (mixing stability) and the outlet (jet stability), and satisfies... .

[0063] (4) Construction of weights for robust optimization sub-objective function Since velocity field stability (related to trajectory accuracy) and pressure field stability (related to flow efficiency) have different focuses in their optimization logic, we define them as independent sub-objective functions. Based on process requirements, a global importance weighting coefficient is introduced. and : Sub-target 1 (Velocity field robustness): (20) Sub-target 2 (pressure field robustness): (twenty one) in, and They represent process parameters respectively Robust optimization sub-objective functions for the velocity field and pressure field are given as independent variables. This represents the comprehensive uncertainty index of the global velocity field; This represents the comprehensive uncertainty index of the overall pressure field. Within this framework... and It not only adjusts the difference in dimensions, but also represents the focus of optimization. For example, when pursuing extremely fine lines, increasing... Forced compression of velocity field fluctuations; when pursuing high-throughput continuous printing, increase To ensure minimal pressure fluctuations.

[0064] (5) Multi-objective optimization vector integration Finally, a mathematical model for the multi-objective robust optimization problem is established. The optimization process aims to find a set of process parameters. (e.g., sheath gas velocity, carrier gas velocity), so that the uncertainties of the velocity and pressure fields reach an optimal balance across the entire domain: (twenty two) This multi-objective optimization function will serve as the label data for training the GPR surrogate model in S3.2. By solving for the Pareto optimal solution, the most "sluggish" and efficient process parameters in the face of random disturbances will be identified.

[0065] S3.2. Global Optimization Based on GPR Proxy Model and Multi-Objective Optimization Algorithm This stage will transform the physical parameter space into a computational space through the evolutionary logic of a genetic algorithm. Combined with... Figure 13 The parameter encoding mechanism and Figure 14 The optimized process, detailed steps are as follows: (1) Binary encoding of decision variables and population initialization To adapt to the search mechanism of the NSGA-III algorithm, the physical process parameters must first be digitized. For example... Figure 13 As shown, the study selected sheath gas flow rate (SHGFR) and carrier gas flow rate (CGFR) as decision variables, and used binary encoding to convert them into chromosome sequences: Encoding mapping: Mapping continuous flow velocity ranges to a length of... binary string and .

[0066] Physical transformation: During algorithm execution, binary entities are converted back to physical parameter values ​​using linear interpolation formulas. (twenty three) Where D (code) represents the decimal integer value corresponding to the code. The initial population generated from this ( Figure 14 The top section forms the starting point for global optimization.

[0067] (2) Parallel fitness prediction driven by surrogate model After the algorithm generates a new individual, the system decodes it and inputs it into the pre-trained GPR model 1 (velocity field uncertainty) and GPR model 2 (pressure field uncertainty). For example... Figure 14 As shown in the dashed box, this stage achieves efficient parallel evaluation: Target prediction: The model directly outputs the area-weighted standard deviation of the velocity field under the corresponding process combination. With the area-weighted standard deviation of the pressure field .

[0068] Adaptive evaluation: Based on the multi-objective function established in S3.1, calculate the fitness vector for each individual: (twenty four) This process extracts only the predicted mean of GPR as the input for evaluating the multi-objective optimization function, thereby greatly reducing the time cost caused by complex flow field simulation.

[0069] (3) NSGA-III evolutionary iteration based on reference point Iterative optimization of the population using the NSGA-III algorithm ( Figure 14 (Left-side loop). This process simulates natural selection using the following operators: Selection: Retain superior individuals based on their non-dominant ranking.

[0070] Crossover and mutation: Gene exchange and mutation are performed at the binary bit string level to maintain population diversity and ensure that the search can escape local optima.

[0071] Reference point mechanism: Due to the often mutually restrictive and competitive relationship between velocity field uncertainty (objective one) and pressure field uncertainty (objective two) in complex fluid dynamic coupling mechanisms (i.e., it is difficult to simultaneously reach the absolute minimum under the same process parameters), after multiple generations of crossover, mutation, and selection evolution, the NSGA-III algorithm, upon reaching the convergence condition, does not output a single absolute optimal solution, but rather a set of non-dominant Pareto equilibrium points. This set of points constitutes a uniformly distributed Pareto front in the target space, and each equilibrium point in the set represents a combination of process parameters that balances velocity field stability and pressure field stability. Unlike conventional algorithms, NSGA-III uses reference points to maintain the uniformity of the Pareto solution set, ensuring that a widely distributed trade-off is found between velocity field and pressure field stability.

[0072] S3.3. Pareto Optimal Boundary Identification in Global Robust Process Space (1) Mathematical definition of the Pareto optimal solution set in the target space After the aforementioned multi-objective global optimization iterations, the NSGA-III algorithm finally converges and outputs a set of discrete Pareto Equilibrium Points. These equilibrium points, obtained through optimization, constitute the target space. Figure 15 The non-dominated solution set boundary is shown as 'a'. Each equilibrium point in the set represents an optimal configuration of process parameters that cannot be further surpassed under the current fluid dynamic constraints.

[0073] From a mathematical perspective, any equilibrium point in this set of optimization results corresponds to a combination of process parameters. A value is defined as Pareto optimal if and only if it is within the parameter space. There is no other solution. The following dominance relationship must be satisfied: (25) This formula has been rigorously proven. Figure 15 The rationality of the outermost leading-edge envelope delineated by the optimal equilibrium point array in section a. This equilibrium boundary intuitively and quantitatively reveals the stability of the velocity field. With pressure field stability The nonlinear negative correlation (competition) between them means that once the process parameters fall into the set of these equilibrium points, any further improvement in the flow field performance of one side will inevitably have to come at the cost of sacrificing the performance of the other side, thus providing a series of extreme trade-off schemes with extremely clear boundaries for engineering practice.

[0074] (2) Optimal equilibrium point identification based on the curvature maxima method To automatically identify the most robust "red star" points from the Pareto front, we introduce a formula for calculating the curvature of discrete points. For a set of discrete target vectors on the front surface... Its curvature An approximate estimate can be made using the second-order difference method: (26) Satisfy through search The solution allows us to precisely pinpoint Figure 15 The curvature maxima in region a. This point physically represents the "inflection point" of the system response, where the two competing robustness metrics reach an optimal balance of marginal returns.

[0075] (3) Mapping from target space to physical process space After identifying the optimal target point, it needs to be restored to its original state through inverse mapping. Figure 15The physical parameter space is shown in b. Because the GPR surrogate model establishes a continuous mapping relationship... Optimal process parameters The following relationship must be satisfied: (27) in It is a Pareto optimal set. For example... Figure 15 As shown in b, the red star point has a unique physical location in the coordinate system of sheath flow velocity (SHGFR) and carrier flow velocity (CGFR), which signifies that we have successfully transformed the abstract "dual flow field uncertainty equilibrium" into a specific executable process instruction.

[0076] (4) Geometric boundary delineation of robust process space Furthermore, using the curvature maxima point as the core, by identifying clusters of points with similar curvature values ​​around it, we can define a globally robust process window. This window identification not only considers the optimality of individual points but also reflects the stability envelope of the process parameter combination on the physical response surface when facing random perturbations. This leap from "point optimality" to "spatial robustness" provides a quantitative process criterion for high-reliability aerosol printing.

[0077] Finally, robust process decision-making and reliability assessment based on Pareto equilibrium points are presented for this technology. (1) Verification of flow field uncertainty suppression at the numerical simulation level Figure 16 The distribution cloud map of flow field uncertainty generated based on robust decision parameters is shown. By comparing the uncertainty of the velocity field (a1-d1) and the uncertainty of the pressure field (a2-d2) after uncertainty quantification analysis, it can be seen that: Global fluctuation suppression: Under the identified optimal equilibrium parameters, the amplitudes of the standard deviations of velocity and pressure across the entire flow field are compressed to extremely low levels. Especially in key response regions such as the nozzle outlet (d1, d2), the spatial distribution of uncertainty is extremely uniform, effectively avoiding flow deflection caused by sudden pressure changes or excessive velocity gradients.

[0078] Dynamic response consistency: Compared to before optimization, the area-weighted standard deviation of the key functional area is significantly reduced. This means that even with some airflow disturbance at the input, the physical flow response has entered the "low-sensitivity zone," verifying the scientific validity of the identified red star point as the optimal robust decision point from a numerical simulation perspective.

[0079] (2) Verification of process consistency at the physical manufacturing level To further evaluate the effectiveness of robust decision-making in actual production, this study compared aerosol-printed samples with optimized parameters and original empirical parameters. Figure 17 ): Robust process sample ( Figure 17 a1-a5: Printing parameters identified based on the curvature maxima method ( Figure 17 The samples (a1-a5) exhibited extremely high consistency and morphological stability. The droplet deposition boundaries were clear and the roundness was regular, with almost no statistically significant differences between the multiple sample groups. This demonstrates that robust optimization successfully eliminated the random ambiguity effect caused by polydispersity coupling, achieving high-fidelity manufacturing output.

[0080] Original parameter samples ( Figure 17 b1-b5: In contrast, the printing results based on the original process parameters ( Figure 17 (b1-b5) exhibits severe instability and unpredictability. Numerous satellite droplet scattering occurs at the sample edges, resulting in drastic distortion of the sedimentary morphology and extremely poor consistency among sample groups. This phenomenon directly reflects the vulnerability of traditional "point-optimal" parameter tuning methods to random perturbations.

[0081] (3) Reliability assessment and industrialization significance pass Figure 16 and Figure 17 The joint closed-loop verification yields the following reliability conclusions: Physical logic closed loop: Reduction of flow field uncertainty in simulation ( Figure 16 ) and the improvement in printing quality during the experiment ( Figure 17 The high degree of logical consistency between a1-a5 confirms that the "uncertainty equilibrium" criterion proposed in this study is key to improving the stability of aerosol printing processes.

[0082] Engineering reliability assurance: Decisions based on Pareto equilibrium significantly broaden the tolerance of the process, enabling the manufacturing process to maintain high yield even under environmental fluctuations.

[0083] Example 2 This invention also provides a flow field uncertainty correlation mapping and robust optimization system for high-consistency aerosol printing, mainly comprising four components: a simulation model module, a surrogate model module, and a parameter optimization module. Specifically: The simulation model module is used to establish an aerosol printing flow field simulation model based on the original geometry of the aerosol printing nozzle. Then, based on this simulation model and incorporating uncertainty parameters, a flow field uncertainty spatial mapping dataset is obtained. In this embodiment, the simulation model module is further divided into four functional sub-units: a model building unit, a sampling unit, an orthogonal basis function unit, and a dataset unit. The model building unit is used to establish an aerosol printing flow field simulation model for the aerosol printing nozzle and perform physical simulation analysis. The uncertainty quantification parameters include sheath gas velocity and carrier gas velocity. The sampling unit is used to sample the quantized uncertainty quantification parameters in a random parameter space based on a dimensionality reduction sampling strategy. Based on the established flow field simulation model, it extracts the steady-state physical field response data corresponding to each set of input parameters sampled in the random parameter space. The orthogonal basis function unit is used to construct orthogonal basis functions based on the polynomial chaotic expansion of the aerosol printing flow field. The dataset unit is used to extract a flow field uncertainty spatial mapping dataset covering the nozzle inlet, junction cavity, and outlet regions based on the steady-state physical field response data and the orthogonal basis functions.

[0084] The proxy model module is used to construct a flow field inference proxy model based on the flow field uncertainty spatial mapping dataset and the Gaussian process regression model. In this embodiment, the proxy model module is further divided into four functional sub-units: a coupling unit, a Gaussian model unit, a Gaussian training unit, and an encapsulation unit. The coupling unit performs Latin hypercube sampling within the process parameter space defined by carrier gas flow rate and sheath gas flow rate, and couples it with the geometric space coordinates of the CFD module to construct a four-dimensional feature dataset containing both process and spatial dimensions. The Gaussian model unit constructs a Gaussian process regression model based on the Matern kernel function. The Gaussian training unit performs hyperparameter optimization and training on the Gaussian process regression model based on the four-dimensional feature dataset. The encapsulation unit establishes the flow field inference proxy model based on the hyperparameter-optimized and trained Gaussian process regression model.

[0085] The parameter optimization module is used to construct a multi-objective optimization function based on velocity-pressure field stability, and iteratively solves it using a flow field inference proxy model. Finally, the optimal process parameters for aerosol printing are obtained through Pareto optimal boundary identification. In this embodiment, the parameter optimization module is further divided into three functional sub-units: an objective function unit, an iterative optimization unit, and a Pareto optimization unit. The objective function unit constructs the multi-objective optimization function based on velocity-pressure field stability. The iterative optimization unit uses the multi-objective optimization function as label data for training the flow field inference proxy model. After digitizing the physical process parameters, it performs iterative optimization using a genetic algorithm based on the multi-objective optimization algorithm. The Pareto optimization unit uses the Pareto equilibrium point obtained from the iterative optimization of the genetic algorithm, identifies the most robust inflection point using Pareto optimal boundary identification, and then obtains the optimal process parameters for aerosol printing through mapping from the objective space to the physical process space.

[0086] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made to the technical solutions of the present invention by those skilled in the art without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A flow field uncertainty correlation mapping and robust optimization method for high-consistency aerosol printing, characterized in that, Includes the following steps: Step S1. Based on the original geometry of the aerosol printing nozzle, establish an aerosol printing flow field simulation model. Then, based on the aerosol printing flow field simulation model and combined with uncertainty parameters, obtain the flow field uncertainty space mapping dataset. Step S2. Based on the flow field uncertainty spatial mapping dataset and the Gaussian process regression model, construct a flow field inference proxy model; Step S3. Construct a multi-objective optimization function based on velocity-pressure field stability, and use the flow field reasoning proxy model to perform iterative solution. Finally, obtain the optimal process parameters for aerosol printing through Pareto optimal boundary identification.

2. The method according to claim 1, characterized in that, Step S1 includes: S1.

1. For aerosol printing nozzles, establish an aerosol printing flow field simulation model and conduct physical simulation analysis; among which, the uncertainty quantification parameters include sheath flow velocity and carrier flow velocity; S1.

2. Based on the dimensionality reduction sampling strategy, the uncertain quantization parameters defined by quantization are sampled in the random parameter space; and based on the established flow field simulation model, the steady-state response data of the physical field corresponding to each set of input parameters sampled in the random parameter space are extracted. S1.

3. Construct orthogonal basis functions based on the polynomial chaotic expansion of the aerosol printing flow field; S1.

4. Based on the physical field steady-state response data and orthogonal basis functions, extract the flow field uncertainty spatial mapping dataset covering the nozzle inlet, junction cavity, and outlet regions.

3. The method according to claim 2, characterized in that, Step S2 includes: S2.

1. Within the process parameter space defined by the carrier gas flow rate and sheath gas flow rate, perform Latin hypercube sampling and couple it with the geometric space coordinates of the CFD module to construct a four-dimensional feature dataset containing process and spatial dimensions. S2.

2. Construct a Gaussian process regression model based on the Matern kernel function; S2.

3. Based on the four-dimensional feature dataset, perform hyperparameter optimization and training on the Gaussian process regression model; S2.

4. Based on the Gaussian process regression model optimized and trained by hyperparameters, establish the flow field inference proxy model.

4. The method according to claim 3, characterized in that, Step S3 includes: S3.

1. Construct a multi-objective optimization function based on the stability of the velocity-pressure field; S3.

2. The multi-objective optimization function is used as the label data for training the flow field inference proxy model. After the physical process parameters are digitized, the genetic algorithm is used for iterative optimization based on the multi-objective optimization algorithm. S3.

3. Based on the Pareto equilibrium point obtained by iterative optimization convergence using a genetic algorithm, the most robust inflection point is obtained by identifying the Pareto optimal boundary. Then, the optimal process parameters for aerosol printing are obtained by mapping the target space to the physical process space.

5. A flow field uncertainty correlation mapping and robust optimization system for high-consistency aerosol printing, characterized in that, include: Simulation model module, surrogate model module, and parameter optimization module; The simulation model module is used to establish an aerosol printing flow field simulation model based on the original geometry of the aerosol printing nozzle, and then, based on the aerosol printing flow field simulation model and in combination with uncertainty parameters, obtain a flow field uncertainty space mapping dataset. The proxy model module is used to construct a flow field inference proxy model based on the flow field uncertainty spatial mapping dataset and the Gaussian process regression model; The parameter optimization module is used to construct a multi-objective optimization function based on the stability of the velocity-pressure field, and to perform iterative solution using the flow field inference proxy model. Finally, the optimal process parameters for aerosol printing are obtained through Pareto optimal boundary identification.

6. The system according to claim 5, characterized in that, The simulation model module includes: a model building unit, a sampling unit, an orthogonal basis function unit, and a dataset unit. The model building unit is used to establish an aerosol printing flow field simulation model for the aerosol printing nozzle and to perform physical simulation analysis; wherein, the uncertainty quantification parameters include sheath flow velocity and carrier flow velocity. The sampling unit is used to sample the quantized parameters of uncertainty after quantization in the random parameter space based on the dimensionality reduction sampling strategy; and to extract the physical field steady-state response data corresponding to each set of input parameters sampled in the random parameter space based on the established flow field simulation model. The orthogonal basis function unit is used to construct orthogonal basis functions based on the polynomial chaotic expansion of the aerosol printing flow field; The dataset unit is used to extract a spatial mapping dataset of flow field uncertainty covering the nozzle inlet, junction cavity, and outlet regions based on the physical field steady-state response data and orthogonal basis functions.

7. The system according to claim 6, characterized in that, The proxy model module includes: a coupling unit, a Gaussian model unit, a Gaussian training unit, and an encapsulation unit; The coupling unit is used to perform Latin hypercube sampling in the process parameter space defined by the carrier gas flow rate and the sheath gas flow rate, and to couple with the geometric space coordinates of the CFD module to construct a four-dimensional feature dataset containing process dimension and spatial dimension. The Gaussian model unit is used to construct a Gaussian process regression model based on the Matern kernel function; The Gaussian training unit is used to perform hyperparameter optimization and training on the Gaussian process regression model based on the four-dimensional feature dataset. The encapsulation unit is used to establish the flow field inference proxy model based on the Gaussian process regression model optimized and trained by hyperparameters.

8. The system according to claim 7, characterized in that, The parameter optimization module includes: an objective function unit, an iterative optimization unit, and a Pareto optimization unit; The objective function unit is used to construct a multi-objective optimization function based on velocity-pressure field stability; The iterative optimization unit is used to use the multi-objective optimization function as the label data for training the flow field inference proxy model. After digitizing the physical process parameters, it performs iterative optimization using a genetic algorithm based on the multi-objective optimization algorithm. The Pareto optimization unit is used to find the Pareto equilibrium point obtained by iterative optimization based on the genetic algorithm, identify the most robust inflection point by using the Pareto optimal boundary, and then obtain the optimal process parameters for aerosol printing by mapping the target space to the physical process space.

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