A Collaborative Optimization Method for the Design and Fabrication of Tungsten-Copper Functionally Graded Materials

Abstract

This invention discloses a synergistic optimization method for the design and fabrication of tungsten-copper functionally graded materials (FJCs), belonging to the field of FJC design and fabrication technology. The method obtains the sintering window of each layer and the liquid phase fraction difference between adjacent layers through CALPHAD thermodynamic calculations, using these as constraints. It then combines three optimization objectives—interfacial equivalent stress, sintering shrinkage strain difference, and relative density—for initial gradient design. Next, a sintering shrinkage surrogate model is constructed, and spark plasma sintering is performed according to the designed path. Finally, experimental characterization data are compared with the predicted values ​​of the surrogate model. Based on the residuals, the surrogate model parameters and thermodynamic constraint thresholds are simultaneously corrected, and the data is fed back into the design stage for iterative optimization until convergence. This invention achieves synergistic optimization of the gradient structure and fabrication process, ensuring the thermodynamic feasibility and engineering fabrication feasibility of the optimized results.

Description

Technical Field

[0001] This invention relates to the field of tungsten-copper functional graded materials design and preparation technology, and more specifically, to a synergistic optimization method for the design and preparation of tungsten-copper functional graded materials. Background Technology

[0002] Tungsten-copper functionally graded materials (W-Cu FGM) are a novel type of composite material whose composition and microstructure exhibit spatial gradients. The high-tungsten content side possesses characteristics such as low thermal expansion coefficient, high strength, and resistance to thermal erosion, while the high-copper content side exhibits high thermal conductivity. An intermediate transition layer effectively mitigates internal thermal stress. This material shows broad application prospects in high-tech fields such as plasma-facing components in nuclear fusion reactor divertors, high-power semiconductor heat sinks, electromagnetic railgun rails, and high-voltage electrical contacts.

[0003] Tungsten (W) and copper (Cu) are almost immiscible in both solid and liquid states, and their thermophysical properties differ significantly—tungsten's coefficient of thermal expansion is approximately [missing value]. The coefficient of thermal expansion of copper is approximately This significant thermal expansion mismatch leads to severe interfacial thermal stress problems in W-Cu FGM during fabrication and service. Setting a compositional gradient transition layer between the tungsten and copper ends can effectively alleviate thermal stress concentration. However, how to scientifically design the gradient structure (including the number of gradient layers, the composition of each layer, and the compositional distribution function) and how to match it with a suitable fabrication process have long been core challenges restricting the performance improvement of W-Cu FGM.

[0004] In terms of gradient structure design, numerous studies have employed the finite element method to simulate, analyze, and optimize the thermal stress of W-Cu FGM. Zhu Dahuan et al. used ANSYS software to simulate and analyze the thermal stress of W / Cu FGM on the first wall of a divertor, finding the optimal number of gradient layers to be 5 and the Cu concentration distribution index to be 0.4. Related studies also found that the maximum thermal stress of W / Cu FGM first decreases and then increases with the increase of the composition distribution index, reaching a minimum when the composition distribution index is 1. Goupee and Vel proposed a multi-objective optimization method for functionally graded materials, using a non-dominated sorting genetic algorithm with an elitist strategy to search for the Pareto front, where the second model problem is the optimization of W / Cu alloy FGM under high heat flux. However, the aforementioned studies only focus on optimizing gradient structure parameters (number of layers, composition distribution index, etc.) and do not incorporate fabrication process parameters and gradient structure parameters into a unified optimization framework.

[0005] In terms of fabrication processes, spark plasma sintering (SPS) is widely used in the preparation of W-Cu FGMs due to its advantages such as rapid heating, short sintering time, and near-net-shape forming. Previous studies have used SPS to prepare multilayer W-Cu gradient composites at 900–1000℃. However, the existing SPS process parameters (temperature, pressure, holding time, etc.) for W-Cu FGM preparation are preset values ​​and have not been jointly optimized with the gradient structure design. More importantly, SPS sintering of W-Cu FGMs faces a unique challenge: due to the immiscibility of the W-Cu system, the liquid phase fraction differs significantly between gradient layers with different Cu contents at the sintering temperature—the high-Cu layer exhibits a large amount of liquid phase at lower temperatures, while the low-Cu layer remains in the solid-state sintering stage. Studies have shown that when the Cu content difference between adjacent gradient layers is too large, it leads to interlayer shrinkage mismatch, which in turn causes cracking; under thermal shock conditions, the liquid copper phase in the high-Cu layer may even migrate and seep out along the tungsten framework. However, existing studies have only provided empirical descriptions of this phenomenon (such as "the Cu content of adjacent layers should be similar"), without giving quantitative thermodynamic constraints or incorporating them into the optimization model of gradient design.

[0006] In recent years, machine learning methods have been introduced into the design and fabrication of functionally graded materials (FGMs). Reports have documented the combination of CALPHAD thermodynamic calculations with machine learning and multi-objective optimization for materials design, and some studies have constructed an optimization loop of "finite element simulation - machine learning surrogate model - active learning - verification closed loop." However, the subjects of these studies are three-dimensional woven W-Cu composites or miscible systems (such as Fe-Cr-Co-C alloys), whose mismatch mechanism is completely different from that of W-Cu immiscible layered FGMs. In miscible systems, the difference in sintering behavior between different component layers is mainly reflected in different densification rates, while in W-Cu immiscible systems, the difference between different component layers is reflected in a step change in liquid phase temperature and liquid phase fraction, the latter leading to a more severe interlayer mismatch problem. Therefore, the above methods cannot be directly transferred to the optimization design of W-Cu immiscible layered FGMs.

[0007] In summary, existing technologies suffer from at least the following systemic defects: gradient structure design, optimization of preparation process parameters, and thermodynamic feasibility assessment are disconnected, and a systematic method capable of achieving synergistic optimization across the entire chain of W-Cu FGM—"design-thermodynamic constraints-preparation-feedback correction"—has not yet been formed. Specifically: First, at the structural design level, existing optimization methods only target mechanical properties such as thermal stress, failing to incorporate SPS preparation process parameters (temperature, pressure, holding time, etc.) as design variables into the optimization framework; second, at the preparation level, SPS process parameters are all preset and not jointly optimized with gradient structure design, and the constraints on preparation feasibility caused by the unique interlayer liquid phase fraction jumps and sintering windows of the W-Cu immiscible system are not considered; third, at the optimization method level, a few studies introducing machine learning or closed-loop iteration focus on three-dimensional woven structures or miscible systems, failing to specifically design for the thermodynamic characteristics of W-Cu immiscible layered FGMs, and failing to form a complete closed loop of "experimental verification-data feedback-synchronous correction of model and constraints-re-iteration." Summary of the Invention

[0008] To address the shortcomings of existing technologies, this invention provides a synergistic optimization method for the design and fabrication of tungsten-copper functionally graded materials (FGMs). This method incorporates CALPHAD thermodynamic calculations into the gradient design, using the difference in liquid phase fraction between adjacent gradient layers and the sintering window as thermodynamic feasibility constraints. It employs interfacial equivalent stress, the difference in sintering shrinkage strain between adjacent layers, and the relative density of each layer as joint optimization objectives. By combining a sintering shrinkage surrogate model and experimental verification using spark plasma sintering, a complete closed-loop iterative optimization process of "thermodynamic constraints - surrogate prediction - experimental verification - bidirectional feedback correction" is formed, thereby achieving synergistic optimization of the W-Cu FGM gradient structure and the fabrication process.

[0009] To achieve the above-mentioned objectives, the present invention provides the following technical solution:

[0010] A method for the synergistic optimization of the design and fabrication of tungsten-copper functionally graded materials includes the following steps:

[0011] Step (1): Thermodynamic calculations

[0012] Based on the CALPHAD (CALculation of PHAse Diagrams) method, a target composition range is formed according to the end-face composition determined by the service conditions. The W-Cu system thermodynamic database is then used to calculate the liquidus temperature at each Cu content point within the target composition range. solidus temperature and liquid phase fraction-temperature relationship To obtain the sintering window of each layer and the difference in liquid phase fraction between any two adjacent layers .

[0013] The W-Cu system is a strongly immiscible system, almost immiscible in both solid and liquid states. During SPS sintering, gradient layers with different Cu contents exhibit significant differences in liquid phase fraction at the same sintering temperature: high-Cu layers show a large amount of liquid phase at lower temperatures, while low-Cu layers remain in the solid-state sintering stage. When the difference in liquid phase fraction between adjacent layers... When a certain threshold is exceeded, the capillary force of the liquid phase will drive the molten copper to migrate along the pore channels from the high Cu layer to the low Cu layer, leading to disruption of the compositional distribution and inducing interfacial cracks. Simultaneously, if the sintering window of a certain gradient layer... If the temperature range is too narrow (i.e., the liquidus and solidus lines are too close), the controllable temperature range for this layer during SPS sintering is extremely small, making it prone to over- or under-sintering due to temperature fluctuations. Therefore, and It is a key indicator for evaluating the thermodynamic feasibility of W-Cu FGM gradient structures.

[0014] Step (2): Initial gradient design with preparation constraints

[0015] Using the thermal load spectrum of tungsten-copper functionally graded materials under service conditions as input, the steady-state temperature field and thermal stress field are calculated through finite element simulation, with interface equivalent stress as the main factor. Minimum, adjacent layer sintering shrinkage strain difference Minimum and relative density of each layer The maximum number of gradient layers is determined for the target. and Cu volume fraction in each layer and the result obtained in step (1) and Embedded as a constraint condition in a multi-objective optimization function, the constraint condition is: and .

[0016] Preferably, the multi-objective optimization function in step (2) is in the form of:

[0017]

[0018] in This represents the minimum relative density of each layer. , , These are the weighting coefficients.

[0019] Preferably, in step (2), the Cu volume fraction of each layer is distributed according to a power law along the thickness direction:

[0020]

[0021] in The distance from the tungsten end, This represents the total thickness of the gradient layer. The distribution index, Number of gradient layers It has 4 to 8 floors.

[0022] Furthermore, step (2) also includes the following gradient structure geometric constraints:

[0023]

[0024] and

[0025] in For the first Layer thickness, .

[0026] Furthermore, before step (2), the following is also included: performing a finite element thermal stress pre-evaluation of the initial design parameters of the gradient structure to assess the interface equivalent stress. As an evaluation criterion, if If the target threshold is exceeded, adjust the number of gradient layers. Component distribution index or total thickness of gradient layer Redesign.

[0027] Step (3): Constructing a sintering shrinkage surrogate model

[0028] Cu volume fraction in each layer Spark plasma sintering temperature ,pressure Insulation time As input variables, with relative density and / or linear shrinkage rate Build a proxy model for the output variables. .

[0029] Preferably, the surrogate model is a Gaussian Process Regression (GPR), a Bayesian Neural Network (BNN), or a Physics-Informed Neural Network (PINN). When using PINN, its loss function includes data loss, residuals of the heat conduction equation, residuals of the thermoelastic equation, and boundary condition loss.

[0030] and The constraints in step (2) indirectly affect the formulation and sintering feasible region of each layer, thus the proxy model... by , , , The main input variables are sufficient to cover the dominant factors of SPS shrinkage and density. and Pre-filtration conditions, as the feasible domain of the formulation, are independent of run.

[0031] Step (4): Preparation by spark plasma sintering

[0032] Following the gradient path designed in step (2), W powder and Cu powder are separated into layers. The powder is mixed in proportion, stacked and molded, and then formed into a tungsten-copper functional graded material bulk in one step by spark plasma sintering. The process parameters for spark plasma sintering are: sintering temperature 900-1000℃, pressure 30-50MPa, holding time 5-10min, and protective atmosphere of hydrogen or vacuum.

[0033] Preferably, after the discharge plasma sintering in step (4), the process also includes hot isostatic pressing (HIP) post-treatment: temperature 800-1000℃, pressure 120-200MPa, holding for 1-3 hours, and the protective atmosphere is hydrogen or vacuum.

[0034] Step (5): Characterization and Feedback Correction

[0035] The linear shrinkage rate of each layer of the block obtained in step (4) is calculated. Relative density And the characterization of the interface microstructure, combining the measured data with the surrogate model in step (3). The residuals are obtained by comparing the predicted values, and the surrogate model is then corrected based on the residuals. In the parameters and steps (2) and The constraint threshold is determined, and the corrected proxy model and constraints are fed back to step (2) to iterate the gradient path again. Then, step (4) is returned to prepare the model again until the convergence criterion is met. Finally, the gradient component curve and the corresponding discharge plasma sintering process parameter package are output.

[0036] The convergence criterion is: after two consecutive iterations, the decrease in interface equivalent stress is <5%, or the deviation between the measured value of linear shrinkage and the predicted value of the surrogate model is <1%.

[0037] As a preferred option, the Bayesian update method is used in step (5) to correct the hyperparameters of the surrogate model.

[0038] Compared with the prior art, the present invention has the following beneficial effects:

[0039] 1. This invention is the first to use the CALPHAD method to calculate the sintering window of each gradient layer. and the difference in liquid phase fraction between adjacent layers The multi-objective optimization function is embedded as a constraint condition in the W-CuFGM. Addressing the inherent characteristics of the W-Cu immiscible system—significant differences in liquid phase fraction at SPS sintering temperatures among different Cu content gradient layers—a multi-objective optimization function is established by setting... The constraint threshold effectively prevents copper melt migration and seepage and interface cracking caused by excessive jumps in interlayer liquid phase fraction; by setting The constraint threshold (corresponding to the actual window of temperature control accuracy in SPS equipment) ensures that each gradient layer has a sufficiently adjustable temperature range during sintering. Existing optimization methods only target mechanical properties such as thermal stress, without incorporating thermodynamic feasibility constraints into the optimization model. The optimization results may fail to be obtained during sintering due to excessive differences in the liquid phase fraction between layers. This invention fundamentally solves this problem.

[0040] 2. This invention uses gradient structure parameters (number of layers) Cu volume fraction in each layer Component distribution index Thickness of each layer (etc.) and SPS preparation process parameters (temperature) ,pressure Insulation time The design parameters, process parameters, and material properties are all incorporated into a unified optimization framework, and a mapping relationship between these parameters is established through a surrogate model. Simultaneously, hot isostatic pressing (HIP) post-processing after SPS forming serves as an optional solution, providing process assurance for further improvement of the high W-side density. Compared to the existing technology where gradient structure design and fabrication process parameters are preset and optimized independently, this invention achieves true synergy between design and fabrication.

[0041] 3. This invention establishes a complete closed-loop iterative mechanism based on the traditional unidirectional process of "simulation optimization → preparation → verification": Measured data is obtained through SPS experimental characterization; the measured data is compared with the predicted values ​​of the surrogate model to obtain residuals; based on the residuals, the surrogate model parameters and CALPHAD constraint thresholds are simultaneously corrected; and the corrected model and constraints are fed back into the design stage for re-iteration. This "two-way feedback" mechanism—correcting both the surrogate model (improving prediction accuracy) and the thermodynamic constraint thresholds (dynamically adjusting the design feasible region)—enables the optimization system to simultaneously achieve improvements in "computational efficiency" and "physical accuracy" in each iteration. Existing technologies only correct surrogate model parameters (e.g., only updating model hyperparameters), without addressing the dynamic correction of thermodynamic constraint thresholds; this invention achieves dynamic adjustment of the design feasible region through two-way feedback, resulting in higher iteration efficiency and more reliable convergence results.

[0042] 4. This invention also provides constraints on the rate of change of components in adjacent layers (…). , Minimum thickness constraints for each layer A multi-level manufacturability constraint system, including SPS process parameter feasible domain constraints and optional HIP post-processing. These geometric constraints and thermodynamic feasibility constraints ( , Together, these constitute a complete chain of constraints from "thermodynamic feasibility → geometric realizability → process fabrication," ensuring that the optimization results are not only theoretically optimal but also engineeringally fabricable. Attached Figure Description

[0043] Figure 1 This is a flowchart illustrating the synergistic optimization method for the design and preparation of tungsten-copper functionally graded materials according to the present invention. Detailed Implementation

[0044] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are for illustrative purposes only and are not intended to limit the scope of protection of this invention.

[0045] In this invention, the tungsten end (or hot side) refers to the side of the gradient material closest to the service heat source, where the Cu volume fraction is the lowest and the W volume fraction is the highest (approaching pure W); the copper end (or cold side) refers to the side furthest from the heat source, where the Cu volume fraction is the highest and the W volume fraction is the lowest (approaching pure Cu). The distance from the tungsten end, This represents the total thickness of the gradient layer from the tungsten end to the copper end. Indicates the distance from the tungsten end The volume fraction of Cu at that location.

[0046] Please see Figure 1 This invention provides a method for the design and preparation of tungsten-copper functionally graded materials through synergistic optimization, comprising the following steps: Step (1) CALPHAD thermodynamic calculation—obtaining the sintering window of each gradient layer. and the difference in liquid phase fraction between any two adjacent layers Step (2) Gradient initial design with preparation constraints—using interface equivalent stress Minimum, adjacent layer sintering shrinkage strain difference Minimum and relative density of each layer With the goal of maximizing, and To constrain the number of gradient layers, determine the number of gradient layers. and Cu volume fraction in each layer Step (3) Constructing a sintering shrinkage surrogate model—with SPS temperature ,pressure Insulation time As input, with relative density and linear shrinkage rate For output; Step (4) Discharge plasma sintering preparation - stacking molds according to the designed gradient path, and forming a block in one step by SPS; Step (5) Characterization and feedback correction - comparing the measured data with the predicted values ​​of the surrogate model to obtain the residual, and correcting the surrogate model parameters and CALPHAD constraint thresholds, and feeding back to step (2) to iterate again until the convergence criterion is met, and outputting the final gradient component curve and the corresponding SPS process parameter package.

[0047] The steps described above will be explained in detail below with reference to specific embodiments.

[0048] Example 1: CALPHAD Thermodynamic Calculations

[0049] This embodiment focuses on a tungsten-copper functionally graded material for nuclear fusion divertors. The tungsten end (hot side) is pure tungsten (0% Cu volume fraction), and the copper end (cold side) is the target composition W-40Cu (40% Cu volume fraction). The total thickness of the gradient layer is... Thermo-Calc software (or other CALPHAD software such as Pandat or OpenCalphad) was used, and the TCS Cu-based Alloys Database (TCCU) W-Cu system thermodynamic database was invoked to calculate the liquidus temperature at each Cu content point (Cu volume fraction 0%, 10%, 20%, 30%, 40%) within the target composition range. solidus temperature and liquid phase fraction-temperature relationship .

[0050] The W-Cu system is a strongly immiscible system, exhibiting almost no immiscibility in both solid and liquid states. Calculation results show that:

[0051]

[0052] Sintering windows of each gradient layer all ,satisfy The constraint threshold. The difference in liquid phase fraction between any two adjacent layers. The calculations are as follows at the typical SPS sintering temperature of 950℃:

[0053] Adjacent layers and :

[0054] Adjacent layers and :

[0055] Adjacent layers and :

[0056] The liquid fraction difference between the adjacent layers above all satisfy the following conditions. The constraint threshold indicates that the gradient path is thermodynamically feasible to fabricate. (The adjacent layers...) and of Approaching the upper limit of the threshold of 0.15 indicates that this interlayer is a critical interface for gradient design and needs to be controlled through optimization of SPS process parameters in subsequent steps.

[0057] Furthermore, the Cu volume fraction of each layer is calculated using the power-law distribution formula from step (2): ,Pick , In layering, the distance of each layer from the tungsten end The corresponding thicknesses are 0.3mm, 0.9mm, 1.5mm, 2.1mm, and 2.7mm. The percentages are approximately 6.3%, 22.6%, 41.4%, 61.3%, and 81.2% (volume fraction), respectively. These are the percentages for each layer along this gradient path. All meet Constraints, each adjacent layer All of the above are satisfied after calculation. Constraints.

[0058] like (like ), adjacent layers on the high Cu side ( and )of If it exceeds 0.15, there is a risk of excessively large jumps in the interlayer liquid phase fraction, which may cause copper liquid to migrate and seep out along the tungsten framework from the high Cu layer to the low Cu layer during SPS. (like The low W-side component changes too gradually, leading to reduced gradient layer efficiency and interface equivalent stress. Significantly increased. Therefore, The selection of the layer is based on clear technical criteria. If more precise gradient control is required, the number of layers can be increased to 6 to 8, and the thickness of each layer can be adjusted accordingly to ensure that the total thickness remains constant.

[0059] Example 2: Finite Element Simulation, Gradient Structure Geometric Constraints, and Multi-Objective Optimization

[0060] (I) Pre-assessment of FEM thermal stress

[0061] This embodiment uses the service conditions of a nuclear fusion divertor as input and employs ANSYS finite element software to perform thermo-mechanical coupling simulation of tungsten-copper functionally graded materials. The service conditions are set as follows: surface heat flux density 10 MW / m². 2 The temperature of the tungsten end (hot side) is 800℃, and the temperature of the copper end (cold side) is 150℃.

[0062] First, a finite element thermal stress pre-evaluation is performed on the initial design parameters of the gradient structure: taking the initial design as... layer, , , Calculate the interface equivalent stress according to the power-law distribution. .like If the target threshold is exceeded (set to 300 MPa in this embodiment), then adjust. , or Redesign. Iterative pre-evaluation—recalculation after each parameter adjustment. And compare it with the threshold until the requirement is met—when , , hour If the initial design meets the pre-evaluation threshold requirements, it will proceed to formal multi-objective optimization. If the initial design does not meet the threshold requirements, adjustments will be made first. Value (within the range of 0.8 to 1.6), then adjust. Values ​​(within the range of 4 to 8 floors), adjusted last. Value (in the range of 1 to 5 mm).

[0063] (ii) Geometric constraints of gradient structure

[0064] Step (2) also includes gradient structure geometric constraints: and In this embodiment, , Thickness of each layer Use 0.6mm for all. The distributions are 6.3%, 22.6%, 41.4%, 61.3%, and 81.2%. Adjacent layers... They are respectively: , , , (vol% / mm), all within Within the range (take) (That's all). Thickness of each layer The geometric constraints must be met. If a non-uniform layer thickness is used, the ratio must be checked group by group to ensure it does not exceed the limit. Range; Due to the steep power-law distribution on the high Cu side, it is recommended to appropriately increase the layer thickness or the number of layers. Make a single jump Decrease.

[0065] (III) Multi-objective optimization

[0066] The multi-objective optimization function is in the form of:

[0067]

[0068] in , , For the weighting coefficient, in this embodiment, we take... , , (The weight allocation is determined based on the priority requirements for thermal stress mitigation in the divertor's service scenarios.) The constraints are as follows: and .

[0069] In gradient layers Layers, component distribution index Total thickness of gradient layer Within the joint parameter space, the NSGA-II multi-objective evolutionary algorithm is used to search for the Pareto optimal solution set. The population size is set to 150, the number of generations is 300, the crossover probability is 0.8, and the mutation probability is 0.1. The optimization results show:

[0070] when , , hour, The efficiency was reduced by approximately 60% compared to the case without gradient layers.

[0071] Sintering shrinkage strain difference between adjacent layers The maximum value appears and Between, approximately 0.8%;

[0072] Relative density of each layer The predicted value is approximately 96.5% (SPS-only mode).

[0073] Based on Pareto front analysis, the optimal solution combining the three objectives is: layer, , The Cu volume fractions in each layer follow a power-law distribution, approximately 6.3%, 22.6%, 41.4%, 61.3%, and 81.2% (volume fraction). This gradient design corresponds to... The maximum value is approximately 0.14 (appearing in...). and Between), satisfying the constraint ≤0.15; each layer The minimum value is approximately 140K, satisfying the constraint of ≥30K.

[0074] Finite element simulations also show that after the introduction of the gradient layer, the maximum equivalent stress changes from a concentrated distribution at the tungsten / copper interface to a uniform distribution along the gradient direction, effectively alleviating stress concentration caused by thermal mismatch. Meanwhile, at 10MW / m²... 2 Under surface thermal shock, the surface temperature of the gradient material is below 800℃, demonstrating good thermal shock resistance.

[0075] Example 3: Construction of a Sintering Shrinkage Surrogate Model

[0076] This embodiment constructs a sintering shrinkage proxy model based on the gradient path and SPS process parameter space obtained by optimization in step (2).

[0077] Cu volume fraction in each layer (6.3%, 22.6%, 41.4%, 61.3%, 81.2%), SPS sintering temperature (900~1000℃), pressure (30-50MPa), heat preservation time (5–10 min) was used as the input variable, with relative compactness as the criterion. and linear shrinkage rate For output variables.

[0078] This embodiment preferentially uses Gaussian Process Regression (GPR) as the surrogate model, and the kernel function is a combination of radial basis function (RBF) and white noise kernel. The training dataset is generated as follows: 50 parameter combinations are selected in the input parameter space using the Latin hypercube sampling method, and then... , , , SPS experiments were conducted to measure the corresponding values. and This creates a dataset of "design parameters - process parameters - material properties".

[0079] The model training uses 5-fold cross-validation, and the coefficient of determination on the test set is... The accuracy reached 0.92, with a root mean square error (RMSE) of less than 1.5%. (Surrogate model) The prediction accuracy meets the requirements of subsequent closed-loop iterations.

[0080] and The constraints in step (2) indirectly affect the formulation and sintering feasible region of each layer, thus the proxy model... by , , , The main input variables are sufficient to cover the dominant factors of SPS shrinkage and density. and Pre-filtration conditions, as the feasible domain of the formulation, are independent of run.

[0081] As an alternative, the surrogate model can also employ a Bayesian Neural Network (BNN) or a Physics-Informed Neural Network (PINN). When using PINN, the network structure consists of four fully connected layers (128 neurons per layer), with the input layer corresponding to... , , , Four variables, corresponding to the output layer and Two variables; the activation function is Swish; the loss function includes data loss (mean squared error between predicted and experimental values), residuals of the heat conduction equation (residuals of the steady-state heat conduction equation based on Fourier's law at discrete points), residuals of the thermoelastic equation (residuals of the thermoelastic constitutive equation at discrete points), and boundary condition loss (degree of satisfaction of boundary conditions for temperature and stress fields). PINN is trained using the TensorFlow framework with the Adam optimizer, and the learning rate is from... Gradually decrease to BNN employs variational inference, approximating the posterior distribution by minimizing the ELBO (Evidence Lower Bound) loss function.

[0082] Example 4: Preparation and Characterization by Spark Plasma Sintering

[0083] (a) SPS-only preparation

[0084] Following the gradient path optimized in Example 2 (N=5 layers, Cu volume fractions of each layer being 6.3%, 22.6%, 41.4%, 61.3%, and 81.2%, total thickness t=3mm, and thickness of each layer being 0.6mm), high-purity tungsten powder (purity ≥99.9%, average particle size 2-5μm) and high-purity copper powder (purity ≥99.9%, average particle size 5-10μm) were mixed according to the gradient path (N=5 layers, Cu volume fractions of each layer being 6.3%, 22.6%, 41.4%, 61.3%, and 81.2%, total thickness t=3mm, and thickness of each layer being 0.6mm), respectively. The powders were mixed according to the specified proportions. The mixing was carried out using a three-dimensional mixer under Ar atmosphere protection at a speed of 50 r / min for 2 hours to ensure uniform mixing of each layer of powder.

[0085] The mixed powder is sequentially loaded into a graphite mold using a layer-by-layer powder-laying method. After each layer is laid, a pre-pressure of 5 MPa is applied for pre-compression before laying the next layer. After all layers are laid, the mold is pressed into shape under a pressure of 30–50 MPa.

[0086] A graphite mold containing gradient green bodies was placed in a spark plasma sintering system. After evacuating to ≤10 Pa, hydrogen gas was introduced as a protective atmosphere. The sintering process parameters were: heating rate 100℃ / min, sintering temperature 950℃, pressure 40MPa, and holding time 8min. After sintering, the material was cooled to room temperature in the furnace and demolded to obtain tungsten-copper functionally graded material blocks.

[0087] The linear shrinkage rate of each layer of the SPS sintered bulk material was measured. and relative density The measurement results are as follows:

[0088]

[0089] In SPS-only mode, the relative density of each layer increases with increasing Cu content, with higher density on the W side (layers 1-2) above W85. The density is approximately 94.5%–95.8%, with an average density of approximately 96.1% across the entire layer. The linear shrinkage rate of each layer increases with increasing Cu content, and the maximum shrinkage strain difference between adjacent layers is [not specified]. The concentration was approximately 2.3% (between the first and second layers), which is within the range predicted by the surrogate model. Scanning electron microscopy (SEM) showed that the interfaces of each gradient layer were clearly visible, the interlayer bonding was good, and no macroscopic cracks were found. XRD analysis showed that only W and Cu diffraction peaks were present in each layer, and no other impurity phases were detected.

[0090] (II) Preparation of SPS+HIP

[0091] Based on SPS sintering, some samples underwent hot isostatic pressing (HIP) post-treatment. The HIP process parameters were: temperature 950℃, pressure 150MPa, holding time 2h, and argon atmosphere.

[0092] The relative density of each layer of the sample was significantly improved after HIP:

[0093]

[0094] After HIP, the relative compaction of the entire layer is ≥98.5%, with W85 and above on the higher W side (layers 1-2). The densification rates reached 98.6% and 99.0% respectively, meeting the high-W-side densification requirements. HIP treatment effectively eliminated residual interconnected pores in the SPS-only sample, resulting in more uniform density across layers. SEM observation at the interface showed that the interlayer interfaces were more compact after HIP, and no interlayer slippage or interpermeation due to excessive pressure was observed. This result indicates that the HIP pressure range of 120–200 MPa set in this invention is reasonable—a median pressure of 150 MPa is sufficient to achieve full-layer densification. And it will not damage the layered interface due to excessive pressure.

[0095] As a comparative example, when the HIP pressure is reduced to 100MPa, the W85 and above high W side (layers 1-2) Only about 96.5% to 97.0% was achieved, which is insufficient to reach the target of 98.5%. Therefore, setting the lower limit of HIP pressure to 120 MPa has clear experimental basis.

[0096] Example 5: Characterization Feedback and Closed-Loop Iteration (Main Link)

[0097] The measured data of SPS-only in Example 4 ( , ) and the proxy model in Example 3 The predicted values ​​are compared. The first round of the surrogate model uses a small training set of initial values ​​(20 groups, overall). However, the prediction deviation on the high Cu side is relatively large, with the deviation in linear shrinkage rate of the first and second layers reaching 1.2%. The finite element analysis yielded a value of 282 MPa, which is approximately 2.5% higher than the predicted value of 275 MPa from step (2). The deviation in the first-round linear shrinkage rate is approximately 1.2% (layers 1-2), exceeding the convergence threshold of 1%. The deviation from the predicted values ​​indicates model instability on the high Cu side, triggering feedback correction: Bayesian updates are used to correct the surrogate model hyperparameters (the training set is expanded from 20 to 40 groups, overall...). (The prediction bias on the high Cu side was significantly reduced by increasing the value from 0.82 to 0.89). Meanwhile, [the prediction bias on the high Cu side was significantly reduced]. The constraint threshold was tightened from 0.15 to 0.13 (because...). First round of flow channel with 30% interface The forecast is too low, indicating that this area... (The model becomes unstable when approaching the threshold) The constraint threshold was increased from 30K to 35K (considering SPS temperature control fluctuations), and the corrected surrogate model and constraints were fed back into step (2) for re-optimization. After re-optimization, Fine-tuned from 1.2 to 1.1. The thickness was adjusted from 0.6mm to 0.65mm (total thickness remains 3mm), and each layer... Make appropriate minor adjustments.

[0098] After the second round of SPS preparation, the measured deviation of linear shrinkage of each layer decreased to 0.6%, and the interfacial equivalent stress was reduced. The test result was 273 MPa, a decrease of approximately 3.2% (<5%) compared to the first round of 282 MPa, which met the convergence criterion. The final gradient component curve and SPS process parameter package were then output.

[0099] This process fully demonstrates the bidirectional feedback mechanism of step (5) of the present invention—it both corrects the hyperparameters of the surrogate model and dynamically corrects the CALPHAD constraint threshold, making the second round of optimization safer and more reliable within the design feasible domain.

[0100] Example 6: Closed-loop iterative multi-round verification (initial severe deviation from the scenario)

[0101] To further verify the effectiveness of the bidirectional feedback mechanism of "simultaneously correcting the surrogate model parameters and CALPHAD constraint threshold" in step (5) of the present invention when the initial design deviates significantly, this embodiment uses a set of initial designs that deviate from the optimal region for multiple rounds of closed-loop iteration verification.

[0102] First round (initial design): layer, , (Each layer is 0.6mm thick) The power-law distribution is approximately 3.2%, 15.2%, 33.0%, 54.8%, 77.5%, and 96.1% (volume fraction). CALPHAD verification revealed that adjacent layers... and Between At 950℃, the value is approximately 0.22, exceeding the 0.15 threshold. Following this gradient path for SPS preparation (950℃, 40MPa, 8min), significant copper melt exudation occurred at the high Cu side interface. Only about 91%.

[0103] Feedback Correction (First Round): The measured data were compared with the predicted values ​​of the surrogate model, and the residuals were relatively large (the deviation in linear shrinkage rate was approximately 2.8%). Bayesian updates were used to correct the hyperparameters of the surrogate model. (Increased from 0.82 to 0.89), while also The constraint threshold was tightened from the initial 0.15 to 0.13, and then fed back into step (2) for re-optimization. After re-optimization... Adjusted to 5 floors , .

[0104] Second round of preparation: SPS preparation is performed again according to the new gradient path. The second round gradient design is as follows: layer, , Thickness of each layer The corresponding thicknesses are 0.65mm, 0.60mm, 0.60mm, 0.60mm, and 0.55mm. The power-law distributions are approximately 5.8%, 20.5%, 39.0%, 58.5%, and 78.0%, respectively. Actual measurements at each layer using SPS-only were also obtained. They are approximately 94.8%, 95.5%, 96.2%, 96.8%, and 97.2%, respectively. The linear shrinkage rate deviation decreased to 0.9%, satisfying the convergence criterion (deviation < 1%), and the iteration terminated. Calculations showed that... and Time Level 1 (W content is relatively high) (Slightly increased), SPS-only Compared to Example 4 The first layer showed a slight increase of 94.5%, which is consistent with the numerical values.

[0105] Comparative analysis: If only the surrogate model parameters are corrected without adjusting the CALPHAD constraint threshold (i.e., the threshold is fixed at 0.15), then after the first round of feedback, it is still possible to produce results near the feasible region boundary. Designs approaching 0.15 pose fabrication risks. This invention actively narrows the feasible design domain by simultaneously adjusting the constraint threshold (tightening it from 0.15 to 0.13), making the second-round design safer and more reliable. This comparison verifies the superiority of this invention's "bidirectional feedback adjustment" (simultaneously adjusting the surrogate model and CALPHAD constraint threshold) compared to the prior art's "adjusting only the surrogate model parameters."

[0106] Example 7: Comparison Experiment of SPS Pressure Parameters

[0107] To verify the basis for selecting the lower limit of SPS pressure of 30MPa in this invention, this embodiment uses the same gradient design ( layer, , SPS was prepared under the same sintering temperature (950℃) and holding time (8min) conditions, and three pressures of 30MPa, 40MPa and 50MPa were used respectively to compare the density and interface integrity under each condition.

[0108]

[0109] The results show that increasing the pressure from 30 MPa to 50 MPa only increases the average density of the entire layer by about 0.5 percentage points (from 95.8% to 96.3%), but at 50 MPa, the high Cu layer ( In the liquid phase emergence stage, 81.2% of Cu liquid infiltrated into adjacent lower Cu layers due to excessive pressure, disrupting the compositional purity of the gradient interface. Although the density was slightly lower at 30 MPa (95.8%), it was close to 96% in the SPS-only mode, and could be increased to over 98.5% after HIP post-treatment (Example 4). Therefore, 30 MPa as the lower pressure limit is a reasonable trade-off between "ensuring sufficient density" and "avoiding interfacial movement and component interpenetration at the layer interface during the liquid phase emergence stage".

[0110] Example 8: Comprehensive Performance Evaluation

[0111] The comprehensive performance of the tungsten-copper functionally graded material prepared by SPS+HIP in Example 4 was tested:

[0112] (1) Relative compaction: The relative compaction of the entire layer is ≥98.5% (Archimedes drainage method). The compaction of each layer is uniformly distributed along the gradient direction, with higher compaction on the W side (layers 1-2) above W85. The Cu content reached 98.6% and 99.0% respectively, with a maximum of 99.5% across all layers (layer 5, Cu volume fraction 81.2%). The HIP treatment effectively eliminated the density gradient, ensuring consistent density throughout the component's interior and on its surface.

[0113] (2) Thermal conductivity: The thermal conductivity at room temperature (25℃) at the W end (first layer, Approximately 160 W / (m·K), at the Cu end (5th layer, The thermal conductivity is approximately 280 W / (m·K), with an arithmetic average thermal conductivity of approximately 180 W / (m·K) for each layer. At 800℃, the thermal conductivity at the W end is approximately 140 W / (m·K), the thermal conductivity at the Cu end is approximately 250 W / (m·K), and the arithmetic average thermal conductivity of each layer is approximately 160 W / (m·K). The thermal conductivity decreases with increasing temperature, which is consistent with the typical temperature dependence of W-Cu materials.

[0114] (3) Interface bonding: SEM observation showed that the interfaces of each gradient layer were clear, without macroscopic cracks or traces of copper liquid seepage. Interlayer shear strength test showed that the interlayer shear strength of each layer was ≥144MPa.

[0115] (4) Thermal shock resistance: The sample was held at 800℃ for 10 min and then quickly immersed in 25℃ water for quenching, and the cycle was repeated 50 times. After thermal shock, there was no cracking or peeling at the sample interface, and no copper liquid seepage from the surface. This result verifies the CALPHAD constraint ( The effectiveness of this method in suppressing copper melt seepage between layers.

[0116] (5) Hardness: The Vickers hardness decreases from about 350 HV at the W end (first layer) to about 120 HV at the Cu end (fifth layer) along the gradient direction, showing a gradient distribution consistent with the composition gradient.

[0117] The above results demonstrate that the tungsten-copper functionally graded materials designed and prepared by the method of this invention achieve synergistic optimization of the gradient structure and preparation process under the constraint of thermodynamic feasibility, resulting in gradient materials with high density, good interfacial bonding, and excellent thermal shock resistance.

[0118] Comparative Example 1: Without Calphad thermodynamic constraints

[0119] To verify the effectiveness of the CALPHAD thermodynamic constraint in this invention, Comparative Example 1 was set up: without CALPHAD constraint (i.e., without setting...). and Under the same finite element thermal stress optimization objective ( ) The minimum gradient design yields the following optimization result: layer, , The percentages are 15%, 50%, and 85%, respectively.

[0120] After verification by CALPHAD, the adjacent layers under this gradient path and Between At the typical SPS sintering temperature of 950℃, the value is approximately 0.28, far exceeding the constraint threshold of 0.15. After SPS preparation following this gradient path (950℃, 40MPa, 8min), the sample... and Significant copper melt seepage and microcracks appeared at the interface. The relative density was only about 92%, far lower than the 96.1% (SPS-only) and ≥98.5% (SPS+HIP) of the optimized sample of this invention. This comparative result shows that without CALPHAD thermodynamic constraint ( and Embedding a multi-objective optimization function, the optimization result that focuses solely on mechanical properties may fail during the preparation process due to excessive jumps in the liquid phase fraction between layers—this is precisely the defect of the prior art pointed out in the background of this invention.

[0121] In addition, in the component distribution index In the comparative experiment: when At that time, the high Cu side ( and ) After SPS preparation, copper melt seeps out from the interface; when At that time, the gradient change on the low W side was too gradual, resulting in interfacial equivalent stress. It reached 342 MPa, exceeding the target threshold of 300 MPa. And when... hour, and Simultaneously satisfying the constraints verifies the present invention. The technical rationality of the range selection.

[0122] Comparative Example 2: Imitating the existing closed-loop agent-only model

[0123] To verify the superiority of the present invention's "simultaneous correction of surrogate model parameters and CALPHAD constraint thresholds" over the prior art's "correction of surrogate model parameters only", comparative example 2 is set up.

[0124] The same initial design as in Example 6 was adopted. layer, , However, in step (5), only Bayesian updates are used to correct the hyperparameters of the surrogate model, without correcting the CALPHAD constraint threshold. The constant value is 0.15. (Constantly 30K).

[0125] After the first round of SPS preparation, Copper melt seeped out at the interface with 96.1% of the surface. The linear shrinkage rate deviation is approximately 2.8%. Feedback only corrects the agent (…). (Upgraded from 0.82 to 0.89), after re-optimization Adjusted to 5 floors , (because The tightening is not complete; the optimizer is still producing outputs at the feasible region boundary. (Design close to the threshold). After the second round of SPS. Slight Cu migration was observed at 50% of the interface. The linear shrinkage rate deviation is approximately 1.3%. Further feedback was received from the repair agent. ), optimizer output , , After the third round of SPS The line shrinkage rate deviation was 0.8% before convergence, requiring a total of 3 rounds of iteration, and the second round still had boundary risks.

[0126] In contrast, embodiment six of the present invention... The value was proactively tightened from 0.15 to 0.13, narrowing the design feasibility region, and the second round of testing yielded positive results. Safety design ( It requires fewer iterations and has lower fabrication risk. A comparison of the two is as follows:

[0127]

[0128] Note: Comparative Example 2, Round 2 Slight Cu migration still occurred at the 50% interface, leaving local porosity. Although the third round was corrected, the porosity was not completely closed under SPS-only conditions, hence the final round... This invention is equivalent to the present invention. In the second round of the present invention, embodiment six achieves a safety design with no boundary risk, and the advantages are even more significant after HIP (≥98.5% vs. approximately 97-98% in comparative example 2).

[0129] The above description is merely a preferred embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Various changes made within the scope of knowledge possessed by those skilled in the art without departing from the concept of the present invention still fall within the scope of protection of the present invention.

Claims

1. A method for the synergistic optimization of the design and preparation of tungsten-copper functionally graded materials, characterized in that, Includes the following steps: (1) Thermodynamic calculation: Based on the CALPHAD method, the target composition range is formed according to the end face composition determined by the service conditions. The W-Cu system thermodynamic database is called to calculate the liquidus temperature of each Cu content point in the target composition range. solidus temperature and liquid phase fraction-temperature relationship To obtain the sintering window of each layer and the difference in liquid phase fraction between any two adjacent layers ; (2) Preliminary design with fabrication constraints: Using the thermal load spectrum of tungsten-copper functional graded materials under service conditions as input, the steady-state temperature field and thermal stress field are calculated by finite element simulation, and the interface equivalent stress is used. Minimum, adjacent layer sintering shrinkage strain difference Minimum and relative density of each layer The maximum number of gradient layers is determined for the target. and Cu volume fraction in each layer and the result obtained in step (1) and Embedded as a constraint condition in a multi-objective optimization function, the constraint condition is: and ; (3) Constructing a sintering shrinkage proxy model: based on the Cu volume fraction of each layer Spark plasma sintering temperature ,pressure Insulation time As input variables, with relative density and / or linear shrinkage rate Build a proxy model for the output variables. ; (4) Preparation by spark plasma sintering: W powder and Cu powder are mixed in layers according to the gradient path designed in step (2). The powder is mixed in proportion, stacked and molded, and then formed into a tungsten-copper functional graded material block by spark plasma sintering in one step. (5) Characterization and feedback correction: The linear shrinkage rate of each layer of the block obtained in step (4) is measured. Relative density And the characterization of the interface microstructure, combining the measured data with the surrogate model in step (3). The residuals are obtained by comparing the predicted values, and the surrogate model is then corrected based on the residuals. In the parameters and steps (2) and The constraint threshold is used to feed back the corrected surrogate model and constraints to step (2) to iterate the gradient path again, and return to step (4) to prepare again until the convergence criterion is met, and output the final gradient component curve and the corresponding discharge plasma sintering process parameter package; the convergence criterion is: the interface equivalent stress reduction is <5% after two consecutive iterations, or the deviation between the measured value of the linear shrinkage rate and the predicted value of the surrogate model is <1%.

2. The method according to claim 1, characterized in that, The CALPHAD calculations in step (1) use Thermo-Calc, Pandat, or OpenCalphad software, along with a W-Cu binary or multi-element thermodynamic database.

3. The method according to claim 1, characterized in that, The multi-objective optimization function mentioned in step (2) is in the following form: in This represents the minimum relative density of each layer. , , These are the weighting coefficients.

4. The method according to claim 1, characterized in that, In step (2), the volume fraction of Cu in each layer is distributed according to a power law along the thickness direction: in The distance from the tungsten end, This represents the total thickness of the gradient layer. The distribution index, Number of gradient layers It has 4 to 8 floors.

5. The method according to claim 1, characterized in that, The process parameters for the discharge plasma sintering in step (4) are: sintering temperature 900-1000℃, pressure 30-50MPa, holding time 5-10min, and protective atmosphere is hydrogen or vacuum.

6. The method according to claim 1, characterized in that, The surrogate model mentioned in step (3) is a Gaussian process regression model, a Bayesian neural network, or a physical information neural network; the loss function of the physical information neural network includes data loss, residual of the heat conduction equation, residual of the thermoelastic equation, and boundary condition loss.

7. The method according to claim 1, characterized in that, In step (5), the Bayesian update method is used to correct the hyperparameters of the surrogate model.

8. The method according to claim 1, characterized in that, Step (4) after spark plasma sintering also includes hot isostatic pressing post-treatment: temperature 800~1000℃, pressure 120~200MPa, heat preservation for 1~3h, and protective atmosphere is hydrogen or vacuum.

9. The method according to claim 1, characterized in that, Step (2) also includes the following gradient structure geometric constraints: and in For the first Layer thickness, .

10. The method according to claim 1, characterized in that, Before step (2), the following steps are also included: performing a finite element thermal stress pre-evaluation of the initial design parameters of the gradient structure to assess the equivalent stress at the interface. As an evaluation criterion, if If the target threshold is exceeded, adjust the number of gradient layers. Component distribution index or total thickness of gradient layer Redesign.

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