Research on the efficient process-oriented structural optimization method of the large-scale vacuum cryostat for fusion reactors

An efficient optimization design for the large and complex components of fusion reactors is crucial to address the engineering design requirements and further promote technical standardization. Based on research status, current engineering designs for fusion reactors have some deficiencies, such as time and energy wastage, inefficiency, and difficulties in covering the typical ‘multi-variable multi-objective’ design requirements. These are pressing and common problems that urgently need to be overcome. To deal with the aforementioned technical challenges, it is vitally important to design an efficient, precise, and normalized approach that is tailored for the development of future fusion reactors. Therefore, this paper proposes a process-oriented optimization design method, which involves Coupled external parameterized modeling, Experimental points design, Response surface optimization, and Structural integrity validation (CERS), to improve the currently inefficient design methods. And the vacuum cryostat, the largest and complex component of a tokamak, is taken as an example to present the basic procedures of CERS. Firstly, the functions, basic structures, load types, analysis methods, and verification criteria of the cryostat are presented in detail. Then, real-time data interaction between external global parametric variables and ANSYS via coupling is established by CERS, which achieves parametric modeling of the cryostat and efficient experimental point design and optimization analysis with multi-variables and multi-objectives in an automatic way. Subsequently, this study demonstrates the significance and sensitivity of various structural parameters of the cryostat from such objectives as maximum deformation, maximum equivalent stress, and total mass. And the optimal set of its structural parameters is obtained by establishing a mathematical optimization model. Finally, the structural integrity is verified. The results indicate that the optimized cryostat maintains a minimum safety margin of 23% and will not suffer fatigue damage under various load events during its service. Moreover, the nonlinear buckling load multiplier ∅ is 5.4, obtained by analyzing the load-displacement curve of the cryostat according to the zero-curvature criterion. This shows that the designed cryostat is stable enough. The proposed method is simple, efficient, and reliable, and can be applied to both the cryostat and other complex components of fusion reactors in engineering design fields. It has great value of practical technical reference and can further promote the standardization of engineering design technology for future fusion reactors.

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Introduction
After more than half a century of development, the tokamak has emerged as the most promising magnetic confinement fusion device that helps to achieve peaceful utilization of nuclear fusion energy and address energy demands and environmental challenges with significant research achievements [1][2][3].A tokamak, as a highly integrated system, contains several large and complex components, such as a vacuum cryostat, vacuum vessel (VV), superconducting magnets, and thermal shields.The reliable design and evaluation of these components are paramount to the safe operation and sustained performance of the entire system.Moreover, the increasing plasma confinement performance leads to the requirement for a tokamak of a larger and more complex size [4], and the operating conditions also become more complicated.Therefore, the engineering design is increasingly difficult.Specifically, more structural parameters should be considered as fusion reactor components grow more complicated.Additionally, the design may involve various requirements, such as structural size, deformation, strength, or mass, rendering it a typical 'multi-variable and multi-objective' process.Currently, engineers mainly use three typical design methods in the optimized design of the components, such as trial-and-error, the control variables, and an orthogonal experimental approach [5][6][7][8][9][10][11][12][13][14][15][16].In the trial-and-error method, structural parameters are manually adjusted to obtain the desirable results based on the designer's experience and intuition.Due to its simplicity, it has been utilized in the design of large and complex fusion device components, such as the VV and cryostat [5,6].Nevertheless, the manual adjustment is both time-consuming and inefficient, with subjective results.The variable-control method is applied to the design and optimization of simple structures, such as magnet coils [7,8], supports [9][10][11], plasma-facing units for divertors [12], and the first wall of the blanket [13][14][15].However, for large and complex structural components, multiple variables should be involved in the design, resulting in a significant increase in experimental points, and further costs in design and time.Although experimental points can be reduced with an orthogonal experimental design, it is extremely complicated to design an orthogonal table with multi-variables for complex components.Therefore, the method is also limited.Basically, traditional design methods are not sufficient enough to justify their comprehensiveness and optimality, failing to cover the requirements of the multi-objectives and high reliability in design.It is worth noting that the development of future commercial fusion reactors relies on a standardized, accurate, and highly efficient design approach, while the aforementioned design methods, such as the trial-and-error method and the variable-control method, are inapplicable to it due to the subjective nature and lack of rigorous theoretical underpinning.Hence, it is necessary and critical to establish a processoriented and standardized engineering design method to overcome the above technical difficulties associated with the complex components of fusion reactors.
To meet the engineering design requirements of future fusion reactors, this paper proposes a process-oriented optimization design method, which involves Coupled external parameterized modeling, Experimental points design, Response surface optimization, and Structural integrity validation (CERS).It can achieve real-time data interaction between external global parametric variables and ANSYS via coupling, which contributes to automated modeling, efficient design and optimization analysis of experimental points, with the response surface methodology (RSM) characterized by multiple variables and objectives for the complex components.Specifically, this method, as an alternative to traditional methods with lengthy processes involving manual adjustment, modeling iteration, and recoupling, can greatly improve the efficiency of the analysis via external parameterized models that are automatically updated and that are driven by the designed experimental parameters.Additionally, the RSM enables the realization that highly precise mathematical optimization models can be constructed with fewer experimental points and that multi-objectives are co-optimized when involving several structural parameters and the influences of their interaction on the objectives, which allows accurate and qualified engineering optimization design.
To further verify the effectiveness of CERS, this article presents a detailed optimization design of the vacuum cryostat, the largest and complex component in the tokamak machine system.Next, in section 2, the functions, general structure, and current engineering design status of the cryostat are elaborated in detail.And the basic load types and combinations of cryostat along with their corresponding criteria are clearly stated for accurate structural integrity analysis and verification.In section 3, the basic procedures of the proposed CERS method are described, and its advantages are presented by Q. Yu et al comparing traditional design methods and currently available optimization techniques.In section 4, a systematic optimization design and structural integrity analysis is performed on the designed cryostat by CERS, which proves accurate and reliable.In section 5, the results are summarized and a general assessment of CERS is provided.

Cryostat function and general structure
The cryostat is derived during the development of a traditional tokamak into a fully superconducting tokamak, as shown in figure 1.As the outermost component of the tokamak, it encloses a VV, superconducting magnet system, and thermal shield, etc.And its major functions include: (1) providing a high vacuum environment (1 × 10 −4 ∼ 1 × 10 −3 Pa) for the superconducting system to limit convection heat transfer and further ensure its stable operation; (2) subjecting to the dead weight (DW) of in-cryostat components and various operational loads; (3) offering interface ports for heating, diagnostics, pumping, cooling, and maintenance; (4) serving as the most important containment barrier to prevent radioactive products from leaking into the environment [17,18].
As shown in figures 1 and 4, limited by the in-cryostat component structure and its functional requirements, the cryostat presents a cylindrical shape whose structure is mainly composed of three parts: a base section, cylinder body, and top lid.The base section is primarily used to support the DW of the incryostat components and transmit loads under various accident conditions.It must possess sufficient strength and stability to ensure the safety of the main machine's structure.The cylinder body provides some interfaces to connect an auxiliary system.And the top lid is usually designed as a detachable component to meet the assembly and maintenance needs of the in-cryostat components.A metal seal between the top lid and the cylinder body is usually pre-tightened by hook bolts to ensure the tight vacuum of the large-diameter flange.

Research status of cryostat design
The research on the vacuum cryostat truly began with the advent of fully superconducting tokamak devices, mainly including the ITER cryostat [19,20], CFETR cryostat [21], EAST cryostat [22], and the JT-60SA cryostat [23,24].Currently, engineering design investigations mostly involve response characteristics like displacement response and stress response, under specific or combined operational conditions, such as normal operation conditions (NO) [25], seismic events [24], ingress of coolant event (ICE) [26], and electromagnetic events (EMs) [27].They use empirical design to determine the final cryostat structure for simulation verification, which may cause excessive or insufficient margins in verification results [28,29].It is time-consuming and inefficient to have another manual iteration of models.To improve the efficiency of cryostat engineering modeling, Ge established a graphics user interface (GUI) for the ITER cryostat via the secondary development of CATIA based on the automation API (application programming interface), which significantly simplified manual operation on the cryostat model [30].However, the developed GUI is limited to parametric modeling of ITER cryostat structures, with lower universality.Furthermore, structural parameter inputs resort to empirical design instead of coupled optimization analysis.
Figure 2(a) clearly shows the limitations of the current design approaches for fusion device components.The current design methods mainly involve two kinds of modes: Modes 1 and 2. The former has two teams, one for design and the other for analysis, while the latter has a single designer undertaking the two tasks.As shown in figure 2, the blue dashed lines of Model 1 in figure 2(a) presents extra procedures compared with Mode 2. In figure 2(a), the initial modeling of fusion device components is completed based on upstream inputs, like site constraints and design indicators of the device, followed by analysis and verification to determine whether the designed structure meets the specified requirements.For multi-objective design of complex components (such as dimensional requirements, deformation requirements, economic considerations reflected by mass, and structural integrity), because of the difficulties involved in cooptimizing various objectives, even the different influences of structural parameters on them, critical variables may be overlooked in the process in which structural parameters are adjusted by experience.It is troublesome to manually adjust structural parameters to cover multi-objective requirements.Additionally, Mode 1 is mostly adopted to design and analyze complex fusion device components, such as the VV and cryostat, that need immense workloads and highly demanding modeling.Any delays in the feedback between the design team and analysis team will incur incorrect design iterations.Therefore, this mode may result in a waste of human resources and inefficient engineering design.

Load types and combinations for the cryostat
To facilitate accurate analysis and verification of the structural integrity of the cryostat, the basic load types and load combinations that it will suffer and the verification standards for related analyses are elaborated in detail below.During its service, the cryostat may be subjected to various loads as follows: (a) Pressure loads mainly include the atmospheric pressure load P(NO) caused by pumping inside the cryostat under normal operation (NO), the pressure P(ICE) on the internal cryostat wall under ICE, and the internal pressure load P(test) under the cryostat test conditions.(b) Inertial loads include the DWs of cryostat and in-cryostat components and the seismic load (mainly including SL-1, SL-2, and SMHV) under seismic events.SL-1 represents an earthquake with a recurrence interval of approximately 100 years, while SMHV is the most severe earthquake that is prone to occur within a time period of approximately 1000 years.SL-2 corresponds to the seismic level required by French nuclear facilities, and it occurs about every 10 4 ∼ 10 6 years [31].The choice of specific level in  generated from enhanced convective heat transfer in the cryostat caused by ICE and the thermal load T(VVBK) transmitted to each cryostat port by the bellows under the vacuum vessel baking conditions (VVBK).Additionally, G10 insulation plates are installed at the connection interfaces between the toroidal field (TF) coils and VV support legs and the pedestal ring of the base section.Therefore, the thermal loads conducted to the pedestal ring can be reduced to the minimum and be neglected under magnetic cooling, VVBK, and NO.
(e) Interface loads are transmitted from the relevant system components (magnets, VV, thermal shield, and bellows) to the cryostat through connecting interfaces, such as the load transmission paths of the ITER cryostat [32].
The cryostat works under various complex loading conditions.Structural safety cannot be effectively evaluated if merely considering a single category of load.It is necessary to classify the load conditions and design combined loads.Based on the failure modes and the probability of various load events, the load conditions of the cryostat can be categorized into four groups, i.e. from Category I to Category IV [32].At the same time, specific level criteria have been established to ensure the appropriate safety assessment for different event categories.Specifically, Category I and Category II load events employ the allowable stress criteria outlined in ASME VIII Division 2 [33], designated as levels A and B, respectively.Absent in ASME VIII, the allowable stress criteria for Category III and Category IV load events are defined in ASME III Div. 1 as levels C and D, respectively [34].It should be noted that seismic load events are regulated in both ASME VIII Division 2 and ASME III Division 1, i.e. in ASME VIII, the SL-1 load event is considered as level C, whereas in ASME III, it falls under level A. Consequently, the more conservative level A is selected for SL-1 in the analysis of the cryostat.Besides, SMHV and SL-2 are classified as Category III and Category IV events, respectively, corresponding to level criteria C and D. It is noteworthy that, as with the level criteria for Category II events of the cryostat, those for Category I events should be selected.The combination load design for cryostat structure strength analysis is usually based on the ASME VIII Div.
2 code [33], as shown in table 1.The comprehensive influences of seismic events and NO condition are considered in the buckling analysis of the cryostat and its combined loads are presented in table 2.

Verification criteria
For the above load analyses, the structural integrity analysis of the cryostat primarily involves the plastic collapse, local failure, buckling, ratcheting, and fatigue analysis.The cryostat structure under different load combinations is analyzed and verified using the ASME VIII Div. 2 code [33] to ascertain its design adequacy.A summary of the verification criteria for various analysis types is given in figure 3. Secondary stresses are not intended to be balanced with external loads but must meet the deformation compatibility equations in elasticity.Their fundamental characteristics are self-limiting, meaning that when secondary stresses reach the yield limit of the material, they only cause localized yielding in the structure of the cryostat while the majority remains elastic.Therefore, the impact of the secondary stresses, for example, those generated by temperature loads, is not considered in buckling and plastic collapse analyses, but in local failure, ratcheting, and fatigue analyses.In addition, it should be noted that ratchet analysis is not considered for Category IV accidents [35].
Eigenvalue buckling and nonlinear buckling analyses are employed to assess structural stability.The former involves two methods: CASE 1 refers to the elastic analysis without considering geometric nonlinearity, and CASE 2 is the elasticplastic analysis with the effects of large deflections.But they can only be used for preliminary assessment.The authenticity and reliability of the calculation results are achieved via multifactor considerations in nonlinear buckling analysis, including material nonlinearity, geometric stiffness nonlinearity, and model imperfections.In nonlinear buckling analysis, the buckling deformation of the cryostat under the first-order mode obtained from CASE 1 is multiplied by a scale factor to produce geometric imperfections whose magnitude is determined by geometric tolerances caused by the manufacturing process of the cryostat.Besides, the nonlinear buckling load multiplier is obtained through the analysis of the load vs displacement curve of the cryostat according to the zero-curvature criterion [36].
The analysis results are subjected to linearized stress or other post-processing, and then verification is performed in accordance with the aforementioned criteria.When the analysis results for all the loading events in table 1 pass, the reliability of the cryostat structure can be confirmed.

The CERS method
For the engineering design of future fusion devices with efficient, accurate, and optimal standards, a comprehensive and systematic approach is required to address the difficulties of optimization design with multi-variables and multi-objectives.Illustrated in figure 2(b), the CERS method proposed in this paper meets these requirements.Firstly, an accurate parameterized model of the complex components of the device is constructed based on upstream design inputs.Meanwhile, all the parameterized variables are coupled with ANSYS.Secondly, the experimental points of RSM are automatically designed and analyzed according to the structure variables that need to be optimized, and the involved objectives.Finally, the optimal structure obtained by co-optimizing multi-objectives is simulated under various accident conditions to verify the reliability of the results.Compared to current design methods, this method fully achieves model iteration, experimental point design, and multi-objective optimization in an automatic way, passing on to computers the bulk of the workload and greatly improving the efficiency of engineering design.Besides, the optimal design can be created based on the regression models of the design objectives and variables constructed by multiobjective optimization, which makes the design of the complex device accurate and reliable.Furthermore, the RSM involved in CERS can establish high-precision response regression models in less time because of fewer experimental points than other optimization algorithms, such as Screening and Multi-Objective Genetic Algorithms (MOGA); therefore, it is especially applicable to the engineering optimization design of large and complex fusion devices.
The following section will introduce the engineering design and optimization of the vacuum cryostat with the CERS method.

Parametric modeling of the cryostat
ITER is the largest international collaborative project whose design draws on the wisdom of the international fusion community.The basic structure of the cryostat is designed in this paper with reference to the ITER cryostat [20].The specific structure of the cryostat is shown in figure 4.
To demonstrate the effectiveness of the proposed method for the optimization design of complex and large components for fusion reactors, the size of the designed cryostat is set to ø18 × 21 m.And the material is SS304/SS304L (the cryostat shell is SS304L; the stiffening ribs and the pedestal ring are SS304).The top lid is an elliptical head, with several T-shaped stiffening ribs arranged in the circumferential and radial directions inside.Meanwhile, a reinforcement plate is set on the inner side of the large-diameter flange to ensure that the flange will not produce large deformation and stress concentration under atmospheric pressure and other loads.Besides, some maintenance ports are designed on the top lid.The cylinder body is a cylindrical shell with vertical and toroidal ribs, which features four layers of ports, including upper, middle, and lower rectangular ports as well as the uppermost feeder port.Both sides of each rectangular port are connected with bellows to provide a passage between the VV and external auxiliary equipment.The base section primarily consists of a bottom elliptical head, base cylinder shell, pedestal ring, horizontal plate, skirt support, and feeder ports.The bottom of the pedestal ring is in frictional contact with 16 spherical bearings distributed evenly in the circumferential direction.The skirt support, including the vertical skirt support (VSS) and the toroidal skirt support (TSS), can limit the upward and horizontal motion of the cryostat, and meet the radial contraction of the cryostat shell under ICE.Inside the bottom elliptical head, there are toroidal and radial rib plates on which an operation platform is placed for assembly purposes.Additionally, each feeder port of the cryostat base section serves as a cantilever, and significant stress concentration is produced at the junction of the base cylinder shell with feeder ports under the various combination loads.Therefore, to remove the stress concentration, toroidal and vertical reinforcing ribs should be welded on the outer wall of the base cylinder shell.
In CERS, parametric modeling of the cryostat is realized via established global parameter variables in the external 3D modeling software.Each structure parameter variable is named DS_XX, for instance, DS_CR Cylinder Thickness for the thickness of the cylinder shell.It is imperative to carefully handle the interrelations and the constraint relationships, such as position constraints among various structural parameters, in order to ensure accurate automated modeling.Exclusive of defined ones, all the other structural dimensions are set as parameterized variables.Finally, the established parameterized model is coupled with ANSYS, from which parameter variables are extracted for the design and analysis of experimental points.

Optimization analysis for the designed cryostat 4.2.1. Analysis assumptions.
In this paper, the unnecessary geometrical features that do not affect the simulation results will be neglected.Since the accuracy of the cryostat model can be guaranteed by this proposed method, its structure does not need to be overly simplified during simulation.The assumptions will be made as follows: (1) Since low stiffness of bellows brings a small transmission force in the cryostat, the acting force on the upper, middle, and lower rectangular ports of the cryostat can be ignored.(2) The lifting structures are removed.
(3) The sealing structures, bolts, nuts, threaded holes, and the pretension forces for bolts are not considered.(4) The forces transmitted from the external feeder lines onto cryostat ports are disregarded.(5) The spherical bearing is simplified to impose friction constraints on the bottom of the pedestal ring.(6) All the welding seams are assumed to be welded continuously.

Simulating pre-process.
If all the structural parameters of the cryostat are simultaneously optimized, too many experimental design points will be introduced, leading to an increase in computational costs.Therefore, the optimization analysis is divided into two cases based on the complexity of the cryostat component structures and their interrelations.
The first is the combination of the top lid and cylinder because they are characterized with relatively simple structures and have significant interaction with each other in the stiffness; the second is the base section, optimized independently.Additionally, considering the distribution of cryostat components, the optimization analysis follows a top-down sequence, i.e. starting with the top lid and cylinder, and then the base section.As described in section 4.1, SS304 and SS304L are applied to the cryostat, and their fundamental properties are described in the relevant literature [37].Regarding boundary conditions, constraints for the top lid and cylinder are applied in the Cartesian coordinate system O-XYZ and constraints for the base section are applied in the cylindrical coordinate system O'-X'θZ', as shown in figure 4. For the top lid and cylinder, bonded constraint is set at the flange connection surface.Since the stiffness of the base section is significantly greater than the cylinder, fixed constraint is set at the bottom of the cylinder during analysis.For the base section, the friction coefficient between the pedestal ring and each spherical bearing is 0.2 [38,39], and the bottom of each spherical bearing is set as fixed.Furthermore, since the VSS and TSS limit axial and tangential movements, respectively, in the coordinate system O'-X'θZ', in the skirt bottom, X' is set as free, θ as 0, and Z' as 0. Besides, inertial loads under the corresponding operating conditions should be imposed on the interfaces of the pedestal ring to assess the impact of the in-cryostat components on the base section.As shown in figure 4, the pedestal ring is connected to 16 support legs of the magnet system, with the total mass M 1 preliminarily estimated to be 2750 tons, and 16 support legs of the VV, M 2 2750 tons.The M 1 and M 2 have been appropriately scaled up by a factor of 1.2 to account for the impact of transmitted loads from in-cryostat components under accident conditions.
Normal operating conditions (P(NO) + DW, where P(NO) = 0.1 MPa) are selected to optimize the components in this study.First, the simulation of accident conditions such as electromagnetic and seismic events takes more time because multiple physical modules need to be coupled, causing lower optimizing efficiency.Second, NO, as a Category I event, happens frequently and has a prolonged impact on the cryostat, while accidents like earthquakes are sudden occurrences and leave localized or transient effects on it.Therefore, the overall performance of the cryostat will not be significantly affected under such accident conditions during the optimization process.For instance, the impact of NO and SL-2 as a category IV event on the cryostat structure with a shell thickness of 50 mm are compared, as shown in figure 5.And the results show that the structural deformation response and the maximum equivalent stress under NO are 7.2 mm and 94 MPa, both significantly greater than that under SL-2, with 4.4 mm and 59 MPa, respectively.This indicates that the structural response of the cryostat is primarily influenced by the former event when NO and SL-2 are combined for the analysis.
The optimization simulation method involves linear elastic analysis.Enough margins are considered in optimization objectives for the impact of extreme loads on the cryostat.Namely, the maximum equivalent stress S max of each optimization analysis case under NO is less than S m /1.3.All the cryostat components are meshed by Solid186 elements, with a global mesh size of 80 mm, which ensures that the deviation of results remains below 5%.

Experimental points design with RSM.
The core optimization algorithm of CERS is RSM, and its advantages in addressing multi-variable and multi-objective optimization problems have been elaborated in section 3.For experimental points design of RSM, a Box-Behnken design (BBD) can use fewer experimental points than a central composite design (CCD) to establish the response surface with equivalent accuracy; obviously, it is a more efficient method, especially applicable to the structural optimization design of large complex components with many variables.Therefore, BBD is employed to design the experimental points of RSM for the structural variables of the cryostat.These key structural variables mainly include the thickness of shells, plates, ribs, and the height of stiffeners and elliptical heads, as well as the width of plates.For the involved structural parameter variables of the cryostat, please refer to tables 3 and 4.

Optimization of the top lid and cylinder.
Each key structural parameter of the top lid and cylinder body is considered to comprehensively investigate the impact of the cryostat structure on various optimization objectives.As shown in table 3, 11 optimization variables are identified for the top lid and cylinder, while the remaining structural parameters, constrained by external design input such as site and other system interfaces, are treated as constants.The range for each optimization variable is reasonably determined so that the potential optimal solution is included within it.Meanwhile, it is essential to avoid too wide a range that will generate uncoordinated optimization sets.
Considering the safety and economic aspects of the cryostat, the maximum deformation, maximum equivalent stress, and total mass of the top lid and cylinder are selected as optimization objectives for experimental points design.Of all 188 sets of experiments, 12 sets of repeated experiments at the center point are included, to further assess the magnitude of experimental errors and improve the effectiveness of the experimental design.
Quadratic polynomial models, denoted as G i = f i (A, B, . . ., K) , i = 1, 2, 3, are used to fit the analysis results of the experimental points for the optimization objectives of the top lid and cylinder.And a variance analysis is performed on each fitted model.The results show that the regression models (G 1 , G 2 , and G 3 ) for all the optimization objectives have correlation coefficients greater than 0.95, which demonstrates the models fit well with the analysis results of the experimental points and can effectively predict the relationships between optimization variables and objective responses.Meanwhile, the models are also credible with the signal-to-noise ratios greater than 4.
Based on the aforementioned regression models, the significance and sensitivity of each optimization variable from the top lid and cylinder on the optimization objectives are analyzed, as shown in figure 6.They are obtained by the automatic calculation of the regression mathematical model performed in built-in algorithms of RSM.Pie charts represent the percentage of significance of each variable, and bar charts illustrate their influence trends on optimization objectives (positive: a positive correlation; negative: a negative correlation).For the total mass of the top lid and cylinder, the significance follows the sequence where G has a non-significant influence, while the others are highly significant, with a maximum contribution percentage of 52.83%.Furthermore, all significant variables exhibit a positive correlation effect on the total mass, indicating that the mass of the top lid and cylinder increases with its optimization variables.For the total deformation, the significance follows the order of with C and G having non-significant effects, while the rest are highly significant with a maximum contribution percentage of 29.67%.All significant variables show a negative sensitivity influence, which suggests that an increase in their parameters results in a decrease in the maximum deformation of the top lid and cylinder, whose decrease level is determined by their significance.
Regarding the maximum equivalent stress, the significance follows the sequence where G and K have non-significant influences, while the others are highly significant with a maximum contribution percentage of 32.4%.In addition, all significant variables have a negative sensitivity, showing that the parameters produce a reduction in the maximum equivalent stress of the components.
The influence of each variable on the optimization objectives varies, so does the interaction between them.Take the example of the influence produced by the interaction between the highly significant variables A and H on the objectives of maximum deformation and maximum equivalent stress, as shown in figure 7.As A and H gradually increase, the response surface for maximum deformation declines gradually, with the trend becoming slower.With regard to the response surface of the maximum equivalent stress, along with an increase in A, it initially increases and then decreases with a inflection point within the range of A∈ [30,35].An increase in H leads to a decrease in the response surface with the trend gradually leveling off.
Multi-objective optimization is essentially a process of finding the optimal set of variables based on each response surface.When considering structural safety and cost-effectiveness, the total mass, maximum deformation, and  maximum equivalent stress should be minimized in the top lid and cylinder design.Therefore, the mathematical model for optimization of the structural parameters can be formulated as equation ( 1): In equation ( 1), L_ and U_ represent the lower and upper limits of the optimization variables, respectively.The optimized structural parameter values are rounded up to guarantee good machinability during the actual manufacturing process.Thus, with the RSM, the optimal parameter set for the top lid and cylinder, is obtained as follows:

Base section optimization.
As shown in table 4, 12 optimization variables have been selected for the base section, while the remaining structural features are considered as constant for inputs.
Since the pedestal ring of the base section is classified as SIC-1 (safety importance class), another set of optimization objectives, namely, its maximum deformation, is added to the original optimization objectives based on the overall maximum deformation, maximum equivalent stress, and total mass of the base section.A total of 204 sets of experiments are designed, including 12 sets of repeated experiments at the center point.
The experimental data of each aforementioned optimization objective are respectively fitted with a quadratic polynomial model, G i = f i (L, M, . . ., W) , i = 4, 5, 6, 7. Using the same approach, the results from the variance analyses of regression models (G 4 , G 5 , G 6 , and G 7 ) demonstrate that each fitted model boasts high reliability.
Figure 8 shows the significance and sensitivity of each optimization variable for the various optimization objectives.For the total mass of the base section, the significance from variables is ranked in the order of and all have a highly significant influence with a maximum contribution percentage of 20.29%.Moreover, they exhibit positive sensitivity effects.Regarding the maximum deformation of the base section, the significance of the optimization variables follows the sequence of which T is non-significant and the rest are highly significant with a maximum contribution percentage of 65.13%.Among the highly significant variables, they all present negative correlations with the optimization objective, except R (2.76%) and S (0.44%).Therefore, an increase in R and S results in a slight increase in the maximum deformation of the base section.For its maximum equivalent stress, the significance of the optimization variables follows the sequence N, O, Q, S, and U are non-significant, and the rest are highly significant with a maximum contribution percentage of 43.62%.Additionally, all the variables with high significance have negative correlations with the maximum equivalent stress of the base section.In terms of the maximum deformation of the pedestal ring, the significance of the optimization variables follows the sequence N, O, U, V, and W have non-significant influences.The highly significant variables have negative correlations with the optimization objective, i.e. their increases will leave the maximum deformation of the pedestal ring decreased.
Based on the above analysis, significant variables, P and R, are selected to demonstrate the influence of interactions between variables on the optimization objectives: the maximum deformation and maximum equivalent stress of the base section, and the maximum deformation of the pedestal ring, as shown in figure 9, and the influence on the total mass is obvious, i.e. monotonically increasing.In figure 9(a), as P decreases, the influence trend of R about the maximum deformation of the base section becomes increasingly steep, whereas as R decreases, the influence trend of P about it gradually flattens out.Moreover, with the simultaneous increase in P and R, the maximum deformation of the pedestal ring gradually decreases while the response surface of the maximum equivalent stress first increases and then decreases, with an inflection point within R ∈ [90, 110], as shown in figures 9(b) and (c).In optimization, it is necessary to balance the influences of the optimization variables on the optimization objectives.Meanwhile, when considering structural safety and manufacturing cost-effectiveness, the optimization objectives should be minimized as much as possible.In addition, since the pedestal ring is classified as SIC-1, its maximum deformation is set less than 0.5 mm to prevent excessive deformation from affecting the assembly of in-cryostat components, such as the VV and the magnet system.With the same optimization method introduced in section 4.2.4, the mathematical model for optimization of the base section is formulated as equation ( 2): Finally, the optimal structural parameter set of the base section is obtained using the rounding-up principle:

Verification of the optimization results.
Based on the optimization design results of the top lid, cylinder, and base section, this section focuses on the analysis of their structure strength under NO condition to preliminarily verify the validity and correctness.Figure 10 illustrates the deformation and equivalent stress distribution of each optimized component.For the top lid and cylinder, the maximum deformation at the center maintenance port of the top lid is 7.2 mm, and the maximum equivalent stress S max is 98 MPa, less than S m /1.3 (S m = 138 MPa).The maximum deformation of the base section at the bottom cover plate is 2.5 mm.And the maximum deformation of the pedestal ring under the loads transmitted from the in-cryostat components is only 0.3 mm, which is less than the optimization target of 0.5 mm.Additionally, the maximum equivalent stress of the base section, 77 MPa, covers the design requirements as well.In summary, it is preliminarily proved that the RSM optimization design in CERS is correct and rational from the optimization results that the cryostat components have small deformation, reasonable equivalent stress distribution, and sufficient safety margins.

Structural integrity analysis of the cryostat
To further verify the reliability of the optimization results, it is necessary to analyze the structural integrity of the cryostat under various loading conditions.Since the cryostat design of a specific experimental reactor is not involved in this study, the analysis settings under the various loads and working conditions mentioned in section 2.3 are described in detail in this section by referring to the load conditions and fatigue cycles of the ITER cryostat [40].
(1) Normal operation: the cryostat is operated under a vacuum state, so the external wall is subjected to atmospheric pressure P(NO), i.e. 0.1 MPa.It is also defined that the cryostat will be subjected to vacuum breaks 400 times during its service.(2) VV baking: the pressure load is the same as in NO.Under VV baking, the temperature T(VVBK) of the connecting interfaces of various rectangular ports in the cryostat cylinder conducted from DUCT bellows is initially assessed as 373 K.And the baking cycle is set at 500 times.(3) Test conditions: the cryostat is a large vacuum chamer and an internal pressure test is required.In this case, a gas pressure test is conducted, and the test pressure P(Test) is 0.2 MPa.(4) Helium leakage accident: due to numerous liquid helium cooling pipelines inside the cryostat, any defect in these pipelines will cause helium leakage into the interior of the cryostat, further altering the temperature and pressure of internal walls.Therefore, the most extreme condition Cryostat ICE IV is selected as the load input, i.e. the temperature T(ICE) and pressure of the internal wall P(ICE) are set as 178 K and 0.2 MPa, respectively [26].Only one helium leakage accident is allowed during the cryostat service life; however, for a more conservative assessment, the number of accidents is amplified by a factor of 10. (5) Seismic event: in this study, the SL-2 Design Response Spectra (DRS) of the ITER cryostat, as Category IV events, is used to assess the structural response of the cryostat.The SL-2 DRS can be obtained from [41].And hypothetically, the seismic event will happen five times.
Similarly, the number of the seismic accident is amplified by a factor of 10. (6) EMs: since this paper mainly uses a cryostat as an example to show the CERS method for complex and large components of a fusion device, the exact parameters of magnets and plasma cannot be provided for the analysis of the electromagnetic effects on the cryostat.Furthermore, the related literature research has shown that MDs and VDE loads have little effect on the cryostat [42], which is overshadowed by more extreme accidents.Therefore, the electromagnetic effects on the cryostat are not considered in this paper.
The environment temperature T(NO) during operation is 295.15K and the gravitational acceleration g is 9.81 m s −2 .In addition, the impact of transmitted loads from in-cryostat components is described in section 4.2.2.The simulations involving thermal loads are both based on thermal-structural coupling analysis.And thermal loads applied to the inner and outer walls of the cryostat are set as the third category of the thermal boundary conditions.Specifically, the convection heat transfer coefficient h 1 for the outer wall is taken as 4 W m −2 K −1 under the environment temperature while, for the inner wall, h 2 is 2.3 W m −2 K −1 during the helium leakage accident [43].Other thermal loads are applied as the first category of the thermal boundary conditions, such as T(VVBK).
The seismic analysis begins with pre-stressed modal analysis of the cryostat to obtain its modal shapes of various orders.The number of mode orders is determined by the participation mass coefficient, which is typically greater than 0.85 to guarantee reliable analysis results.Subsequently, ANSYS spectrum analysis is coupled, and the acceleration frequency spectrum of the cryostat is applied in the X, Y, and Z directions respectively within the Cartesian coordinate system O-XYZ for single-point response spectrum analysis.The modal combination method is chosen based on the degree of the correlations between the modal frequencies of the cryostat, i.e. if the modal frequencies are closely spaced, the complete quadratic combination method (CQC) is used; conversely, the square root of the sum of the squares method (SRSS) is applied.The criteria for determining the closeness between adjacent modal frequencies are as follows: if α ⩽ 0.02, ∆ ⩽ 0.1; if α > 0.02, ∆ < 5α; where α represents the critical damping ratio and ∆ is the relative difference between two frequencies.In this paper, the CQC is selected.Furthermore, missing mass effects and rigid response effects are both considered in this analysis.The type of rigid response effect is Gupta, and its starting and ending frequencies are calculated by the spectral peak response frequency f sp and the rigid response frequency f ZPA corresponding to the zero period acceleration (ZPA).Finally, the Newmark's combination rule [44] is used to combine the maximum response results in the X, Y, and Z directions, and the total response of the cryostat under the SL-2 seismic event can be obtained by equation (3) as follows.
where R t is the total seismic response of the cryostat, and R x , R y , and R z are the maximum seismic responses in the X, Y and Z directions respectively.It is worth noting that, in static structural analysis, seismic effects are evaluated by implementing equivalent seismic accelerations.
Since the load cycles of the cryostat are less than 10 4 , defined as low-cycle fatigue, the fatigue damage can be calculated using the strain life curve (E-N curve) [45].According to the form of load action, the cyclic load type is set as zero-based in the fatigue analysis and the fatigue strength coefficient is 1.0.
All the analyses mentioned above are performed based on linear elastic analysis.According to the criteria outlined in section 2.4, the structural strength analysis results of the optimized cryostat are post-processed and summarized in table 5.
It can be seen in table 5 that the VVBK has the minimum safety margin among Category I events, i.e. 23%, which meets the structural strength requirements.Furthermore, for Category IV events, the stress intensity factor k is 2.0; therefore, critical limit stresses corresponding to various analyses under these events are greater than that under NO.By checking all the analysis results, the minimum safety margin of Category IV events is 46%, also greater than that of NO, which indicates that the cryostat structure has great strength to withstand extreme accidents.In fatigue analysis, the fatigue damages for various cyclic loads are significantly less than 1.0.And based on the Miner theory, the calculated total fatigue damage factor is 1.93 × 10 −5 , far less than 1.0 as well.This indicates that the cryostat will not experience fatigue failure during its service life.
Additionally, eigenvalue and nonlinear buckling analyses are performed on the cryostat using the methods described in sections 2.3 and 2.4.The results are presented in figure 11 and  table 6.
Figures 11(a) and (b) present the first-order modal shapes of the cryostat and the corresponding load multipliers, calculated by CASE 1 and CASE 2, respectively, to preliminarily evaluate the structural stability.The calculated load multipliers are 17.7 and 17.6.The geometric tolerance potentially generated by the manufacturing process of the designed cryostat (size: ø18 × 21 m) is preliminary set as ±20 mm.Therefore, in the nonlinear buckling analysis, the first-order modal deformation of the cryostat calculated by CASE 1 is scaled by 20 times to produce geometric imperfections.As shown in figure 11(c), the maximum deformation of the cryostat when buckling is about 85.4 mm.And according to the zero-curvature criterion, the calculated buckling load is about 0.54 MPa, i.e. the load multiplier is 5.4.Basically, the load multipliers obtained from the eigenvalue and nonlinear buckling analyses are both greater than the critical load multipliers, indicating that the designed cryostat structure is sufficiently stable and will not buckle under the combined effects of NO, DW, and SL-2.

Conclusion
In this research, an efficient method, CERS, is proposed to design, optimize, and analyze the largest and complex component of a tokamak, the cryostat.It realizes the automatic parametric modeling through real-time data interaction between external global parametric variables and ANSYS.By considering economy and safety, this paper then selects the total mass, the maximum deformation, and the equivalent stress of the cryostat under NO as optimization objectives.And the experimental point design and optimization analysis with multi-variables and multi-objectives are performed by RSM in CERS.The variance results of the optimization analysis reveal that the mathematical regression models for each optimization objective are significant, precisely demonstrating the sensitivity and significance of the optimization variables for each optimization objective.Furthermore, according to the design requirements of the top lid, cylinder, and base section, the mathematical optimization model for each component is established to obtain the optimal structural parameter set.Then, the optimized components are preliminarily verified.The results suggest that they have enough safety margins to withstand the loads in other bad scenarios since the deformation of each component under normal operating conditions meets the design requirements and its maximum equivalent stress S max is less than S m /1.3.In the last procedure of the CERS method, the structural integrity of the whole cryostat is analyzed to verify its structural strength and

Notes:
a For the cylinder shape reinforced by the stiffening ribs, the β cr is 0.8.b In ASME table 5.5, when considering the combined effects of P(NO), DW, and SL-2, the corresponding load combination is 2.1[P(NO)+ DW]+ 1.7SL-2, with a maximum load factor of 2.1.Therefore, the conservative critical load multiplier ϕ B can be taken as 2.1 for nonlinear buckling analysis.
stability under various accident conditions.The results show that the strength requirements of the optimized cryostat are qualified with the minimum safety margin-23% under the VVBK as Category I events.Additionally, the minimum safety margin under Category IV accidents is 46%, further proving the safety of the cryostat under extreme accident conditions.The total fatigue damage factor is also significantly less than 1 under various cycle loading conditions, which means the cryostat will not experience fatigue failure in service life.To verify the structural stability of the cryostat, eigenvalue buckling analysis (CASE 1 and CASE 2), and nonlinear buckling analysis (CASE 3) are performed.Of all the analyses, CASE 3, which exhibits higher credibility, is applied to assess the structural stability.And its calculated load multiplier is 5.4, which is greater than the critical load multiplier.This indicates that the cryostat is sufficiently stable under external loading conditions.The proposed CERS is being used in the engineering optimization design of the CFETR cryostat now, and its advantages have also been confirmed.In short, this research can greatly improve the efficiency and quality of the complex component design and provide feasible technical guidance for the engineering design of future fusion reactors.

Figure 1 .
Figure 1.The general structural composition and dimensional changes of the fully superconducting tokamak cryostat.

Figure 2 .
Figure 2.Comparison between the current design method and the proposed process-oriented optimization design method for the complex components of a fusion reactor: (a) the current design method; (b) the CERS method.

Figure 3 .
Figure 3. Analysis types and verification criteria for the vacuum cryostat.

Figure 4 .
Figure 4.The parametric design and optimization process of the cryostat structure by the CERS method.

Figure 5 .
Figure 5.Comparison of the impact of NO and SL-2 on the maximum deformation and maximum equivalent stress of the cryostat.

Table 3 .
Optimization variables and their codes and parameter ranges for the top lid and cylinder.elliptical head/E 2000-2400 mm Thickness of top lid toroidal rib/F 30-60 mm Height of top lid toroidal rib/G 550-700 mm Thickness of top lid radial rib/H 30-60 mm Height of top lid radial rib/I 300-450 mm Width of the lower plate of the T-shaped rib/J 250-350 mm Reinforcement plate thickness/K 70-90 mm Note: To avoid more kinds of plate purchase in the manufacturing process of the cryostat, the thickness of the elliptical head shell of the top lid is set the same as that of the cylinder shell.

Figure 6 .
Figure 6.Significance and sensitivity analysis of the influence of the top lid and cylinder structure variables on various optimization objectives.

Figure 7 .
Figure 7.The response surfaces of the interaction effects of significant factors on the optimization objectives for the top lid and cylinder: (a) interaction effects of A and H on the maximum deformation; (b) interaction effects of A and H on the maximum equivalent stress.

Table 4 .
Optimization variables and their codes and parameter ranges for the base section.Component Optimization variables/codes Parameter ranges Illustration Base section Thickness of bottom elliptical head and base cylinder shell/L 20-40 mm Thickness of operation platform/M 20-35 mm Thickness of bottom toroidal rib plate/N 25-40 mm Thickness of bottom radial rib plate/O 25-40 mm Lower plate thickness of pedestal ring/P 90-150 mm Inner plate thickness of pedestal ring/Q 90-120 mm Upper plate thickness of pedestal ring/R 90-150 mm Outer plate thickness of pedestal ring/S 60-90 mm Horizontal plate thickness/T 60-90 mm Thickness of bottom cover plate/U 30-45 mm Thickness of rib plate of base cylinder/V 30-60 mm Height of rib plate of base cylinder/W 150-250 mmNote: To avoid more kinds of plate purchase in the manufacturing process of the cryostat, the thickness of the bottom elliptical head shell is set the same as that of the base cylinder shell.

Figure 8 .
Figure 8. Significance and sensitivity analysis of the influence of the base section structure variables on various optimization objectives.

Figure 9 .
Figure 9.The response surface of the interaction between significant variables R and P for various optimization objectives of the base section: (a) overall maximum deformation; (b) maximum deformation of the pedestal ring; (c) overall maximum equivalent stress.

Figure 10 .
Figure 10.The analysis results of component optimization: (a) deformation distribution of the top lid and cylinder; (b) the equivalent stress distribution of the top lid and cylinder; (c) deformation distribution of the base section; (d) the equivalent stress distribution of the base section.

Figure 11 .
Figure 11.Buckling analyses for the optimized cryostat: (a) eigenvalue buckling analysis for CASE 1; (b) eigenvalue buckling analysis for CASE 2; (c) load vs displacement for nonlinear buckling analysis in CASE 3.

Table 1 .
Load combinations design for cryostat structural strength analysis and the corresponding level criteria.
Note: Each level criteria have a stress intensity factor k. From level A to level D, k represents 1.0, 1.1, 1.2, and 2.0, respectively.

Table 2 .
Load combinations design for buckling analysis of the cryostat.

Table 5 .
A summary of the structural strength analysis results of the optimized cryostat.

Table 6 .
Calculated load multipliers for buckling analyses.