A Comprehensive Study on Fitness Approximation Techniques in Shape Optimization of Aerofoil Forging Preform Tools

Abstract

In the process of complex engineering designs or optimizations, a large number of physical experiments or numerical simulations are required to evaluate certain performance qualities before a satisfactory result can be obtained. In both cases, constructing an approximate model is often necessary to provide a reliable response as an alternative to experiments or simulations. In this paper, three types of approximation models were developed and applied in a shape design of an aerofoil forging preform tool. Their modeling techniques are presented in detail. An optimal Latin hypercube technique was employed for the design of the experiment and sampling with the expected coverage of parameter space. Finite element (FE) simulations of multistep forging processes were implemented to acquire the objective function values for evaluating the forging performance. By a parametric study, the effects of design variables on objective responses and correlations were investigated for a clear insight into their functional nature. Comprehensive analyses and comparisons between different approximate models have been carried out. Finally, an optimization design of a preform tool was successfully achieved based on a particle swarm (PSO) algorithm combined with the proposed approximate model.

1. Introduction

The aeroengine blade is a product which is mainly formed by a hot forging process. It usually possesses a complicated shape, high dimensional accuracy, and superior mechanical performance. Considering the difficulty of blade forging technology, a multistage hot forging process is necessary to make the workpiece gradually approach a desired shape. An appropriate preform design often plays a key role in the forging of aerofoil blades. For a period of time, much attention has been paid to the shape-design of preform or correlative tools in forging optimizations. Substantial progress has been made in the area of multi-objective optimization for preform design through combinations with optimal theories or mathematical programming methods.

Sensitivity analysis as a gradient-based optimization algorithm has been widely used in many engineering fields. In this approach, a sensitivity function about the derivative of the optimal objective with respect to the shape variables should be established first and then calculated through finite element (FE) simulations. The key technique of a continuous sensitivity analysis integrated with the FE method was given in detail by Zabaras and his colleagues. They optimized some forging preforms based on different optimal objectives such as minimizing barreling in the upset of a cylinder [1], controlling grain size, improving the microstructure in the forming process [2], and so on. Compared with continuous sensitivity analysis, a discrete sensitivity analysis is easily applied to actual engineering problems because it does not need to be deeply coupled with numerical simulation code. Using a discrete sensitivity analysis, Gao [3] optimized the initial billet shape of a turbine disk in their consideration of improving microstructure as well as sufficient die-filling and saved material. Zhao [4] established a multi-objective function to optimize preform tools with decreased forging load and improved deformation uniformity. Generally speaking, a sensitivity analysis can get certain positive effects in the area of preform design and optimization. But it is quite difficult to establish a sensitivity function and integrate it with the FE simulation, which limits practical application to a certain degree.

Another possible shape optimization approach for preform design is the topology shape optimization, such as evolutionary structural optimization (ESO). In ESO, the shape of the forging preform is represented by discrete elements. During the iterative process, these elements are added or removed from an initial preform model based on a certain criterion. Then, the preform shape gradually evolves according to this criterion and tends to satisfy some engineering requirements. In recent years, some positive studies have been implemented in the forging of design and optimization based on the topological optimization method by the author [5,6,7]. A breakthrough in terms of the 3D forging optimization is also achieved [8]. At present, the topological optimization method is one of the rare ways to solve the 3D preform optimization problem to produce complicated shapes. However, it is uncertain whether the optimized results are global optimization solutions. The algorithmic rationality and reliability also need to be proved by further studies.

Genetic algorithm (GA), one of the global probabilistic search algorithms, is growing rapidly and has also been widely used in many engineering optimizations. Because of no coupling required between the numerical simulation and the optimization algorithm, the manipulation of the optimization process can be simplified significantly. Roy [9] earlier incorporated a genetic algorithm into the optimal design of a multi-stage forming process. Chung [10] optimized an extruded tool shape using a genetic algorithm. In order to solve the early convergence and inefficiency existing in the standard genetic algorithm, many updated GA-based algorithms have been introduced by improving the original evolutionary rules [11,12]. Guan [13] optimized the H-shape preforming die while pursuing a small grain size and a homogeneous grain distribution of forgings based on the micro-genetic algorithm. Knust [14] suggested an evolutionary algorithm to optimize a preform shape in consideration of the mass distribution and the shape complexity of preform. However, the biggest obstacle for further development of a genetic algorithm is still the lack of efficiency due to frequently executed numerical simulations for objective function evaluation during the optimization process.

In order to reduce the sampling quantities required for using of probabilistic selection algorithms, combined with the mathematical statistics and the numerical analysis, a metamodel (also named surrogate model) technique has been increasingly applied in the optimization iteration for quickly accessing the response of objective function. Trained by the limited sample data, a metamodel can be constructed to predict output objectives, substituting for a time-consuming simulation as long as it has enough approximation accuracy, which could sharply improve the optimization efficiency. At present, the main surrogate models used in shape optimization of forging preform or tool are based on response surface methodology [15,16], the Kriging model [17,18], or neural network [19,20]. Relevant introductions for algorithms can be referred to in the article [21]. However, any surrogate model is only an approximate replacement for the original system function. No matter which model is used, there must be errors between the predicted value and the real value. In consideration of the complexity of engineering problems, comprehensive analyses and comparisons between different surrogate models are very important for actual engineering optimization.

This study mainly focuses on the approximate accuracy when different surrogate models face a multi-objective engineering design optimization with complex nonlinearity. The optimization problem is first formulated as a mathematical modeling. Then, an optimal Latin hypercube sampling (LHS) technique is employed for the sampling plan, which provides uniform coverage of the design domain. A parameter study is implemented and is helpful for uncovering the implicit relationship between design variables and objective functions. Three typical surrogate models are developed to be utilized for predicting the objective responses. Error analyses are carried out to evaluate the reliability of these models in detail. Finally, according to the proposed surrogate model combined with a particle swarm algorithm, a design optimization of an aerofoil preform tool is implemented. The optimal result proves the validation of developed model.

2. Optimization Objectives and Design Variables

2.1. Optimization Objectives

A basic requirement for preform tool design is to ensure a successful preparation for preform which enables sufficient die-filling in the final forging process. So, the die-filling status as a constraint condition was considered to validate the design result:

$$\frac{V_u}{V_t}\le a$$

(1)

where $V_u$ is the unfilled section area, $V_t$ is the desired section area, and a is the fault tolerance, 0.001, in this study.

Increasing deformation uniformity of a forged component is very significant for the achievement of a satisfactory microstructure status with improved mechanical properties. So, one of the objectives for preform optimization is the forging deformation uniformity. It can be described by:

$${\overline{\epsilon }}_{S.D.}=\sqrt{\frac{{\displaystyle \sum _{i=1}^n{\left({\overline{\epsilon }}_{e,i}-{\overline{\epsilon }}_e^a\right)}^2}}{n}}$$

(2)

where ${\overline{\epsilon }}_{S.D.}$ is standard deviation of the effective strain, ${\overline{\epsilon }}_{e,i}$ is the effective strain of element i, ${\overline{\epsilon }}_e^a$ is the average effective strain for all elements, and n is the number of elements. A smaller ${\overline{\epsilon }}_{S.D.}$ means overall increased deformation uniformity.

Due to severe plastic deformation, forging cracks often cause scrapped parts, especially in the forging of aerofoil blades. Reducing the probability of crack emergence is crucial for the improvement of forging quality and reliability. In the present study, the damage behavior of deformed materials was evaluated according to a Cockcroft and Latham ductile fracture criterion [22] described by:

$$D_i={\displaystyle {\int }_{}^{{\overline{\epsilon }}_D}\frac{{\sigma }^{\ast }}{\overline{\sigma }}}d\overline{\epsilon }$$

(3)

where $D_i$ is the local cumulative damage value of element i, ${\overline{\epsilon }}_D$ is the cumulative plastic stain, $\overline{\epsilon }$ is the effective strain of element i, ${\sigma }^{\ast }$ is the maximum principal stress of element i, and $\overline{\sigma }$ is the effective stress of element i. The average of 1% of the elements with the highest damage value as the second objective $D_e^h$ was employed to evaluate the damage degree of deformed material. This objective function can be obtained as follow:

$$D_e^h={\displaystyle \sum _{i=1}^{n/100}{D}_i{V}_i}/{\displaystyle \sum _{i=1}^{n/100}{V}_i}$$

(4)

The shape complexity of preform has significant effects on its preparation. To a close plane figure, shape complexity value C can be expressed by the ratio between the perimeter and the area of the preform profile:

C = L/S

(5)

where S is the area of the preform profile and L is the perimeter of the preform shape. In general, the smaller the C value is, the simpler the preform shape will be. Using this C value, the shape complexity of each preform design can be evaluated.

2.2. Design Variables

A standard forging process of aerofoil blades is illustrated in Figure 1. Concerning the formability and quality of forgings, two forging steps are necessary for relieving cumulative deformations.

Considering the forging stability during the final forging process, the lower die shape of the preform tool was designed to be the same as the final forging tool. The upper die of the preform tool to be optimized is described by a spline curve shown in Figure 2. Nine data points P_j are denoted as design variables to define this spline curve. For decreasing the number of variables, X-axis coordinates of each data point are set as constants. In view of practical engineering, the engineering strain $\epsilon$ expressed as Equation (6) is confined between 0.2 and 0.4 in the deformed preform. Then, Y-axis coordinates of every data point can be confirmed, which gives the optimization range of each variable.

$$\epsilon =(Y_j-Y_j^u)/(Y_j-Y_j^l)$$

(6)

where $Y_j$ is Y-axis coordinate of P_j to be optimized, $Y_j^u$ is the corresponding Y-axis coordinate on upper surface of desired part, $Y_j^l$ is the corresponding Y-axis coordinate on lower surface of desired part, and j is the serial number of data points.

On condition of the plane strain assumption, the initial billet shape was characterized by a circular section with a total area of 105–115% of the desired forging section area for sufficient die-filling. The radius of the circular section r, taken as one of variables, could be optimized within a certain range.

As mentioned above, the shape optimization of preform tool can be formulated as follows:

$$\{\begin{array}{l}\mathrm{Variables}:Y1[44.453,44.799],Y2[44.169,44.791],Y3[44.059,44.968],Y4[43.979,45.049],\\ Y5[43.925,45.042],Y6[43.896,44.955],Y7[43.881,44.770],Y8[43.876,44.489],\\ Y9[44.124,44.466],r[4.936,5.166]\\ \mathrm{Minimize}:{\overline{\epsilon }}_{S.D.},D_e^h\\ \mathrm{subject}\mathrm{to}:V_u/V_t\le a\end{array}$$

3. FE Simulation

The Deform-2D commercial software (v11.3, Scientific Forming Technologies Corporation, Columbus, OH, USA) was utilized for the forging process simulations and the evaluations of objective function. The workpiece material treated as the rigid-viscoplastic model was Inconel718, a constitutive model which can be referred in the literature [23], while all forging tools were assumed as rigid. The spline curve interpolated by the data points P_j was updated during each optimal iteration, and then transformed into the FE geometry of preform tool. For decreasing the error of FE computation, a total of 5000 quadrilateral elements and a consistent meshing style, including remeshing treatment, with 20 rows and 250 columns were strictly applied to the preform model in FE simulation. The initial forging temperature of preform is 1020 °C and the die temperature is 250 °C. A shear friction type was employed for the friction calculation. The friction factor was set to 0.3. A heat transfer coefficient between workpiece and dies is defined to be 11 KW/m²/°C A convection coefficient between workpiece and environment is defined to be 0.02 N/sec/mm/°C. The upper die velocity was 200 mm/s and the low die was stationary in the forging simulation. The FE model for the forging process is illustrated in Figure 1.

4. Parameter Study

An Optimal Latin Hypercube Sampling (OLHS) technique [24] was proposed for the design of the experiment, which can make samples spread as evenly as possible within the space of variable combination. In general, the $(m+1)\times (m+2)/2$ samples at least are necessary for fully covering this space. However, more samples are often needed for highly nonlinear problems. This m is the number of design variables (factors). With consideration of 10 factors (design variables) in this study, 66 samples with their corresponding objective values obtained by FE simulations were determined by using the OLHS method, and another one was created at random for supplement. All 67 initial sample data are shown in

Table 1. Initial sample data.

Number	r	Y1	Y2	Y3	Y4	Y5	Y6	Y7	Y8	Y9	C	${\overline{\mathit{\epsilon }}}_{\mathit{S}.\mathit{D}.}$	${\mathit{D}}_{\mathit{e}}^{\mathit{h}}$
1	5.0103	44.5009	44.5042	44.437	44.111	44.148	44.727	43.949	44.1495	44.145	0.663	0.337	0.0826
2	5.1448	44.6976	44.3319	44.884	45	44.252	44.238	44.428	44.4324	44.2924	0.615	0.442	0.1022
3	5.0811	44.6019	44.7148	44.115	44.078	44.217	44.124	44.141	44.3287	44.4186	0.693	0.419	0.0892
4	5.0705	44.4636	44.7052	44.283	44.061	44.87	44.564	44.647	44.1966	44.2871	0.625	0.365	0.0875
5	4.9466	44.5753	44.2458	44.926	44.654	44.011	44.499	44.483	44.1778	44.2134	0.623	0.388	0.0874
6	5.074	44.788	44.2075	44.674	44.637	44.441	44.173	44.537	43.8949	44.245	0.625	0.395	0.0853
7	5.0917	44.6604	44.6573	44.297	45.033	44.939	44.385	44.592	43.9137	44.2818	0.596	0.352	0.0891
8	4.9643	44.6391	44.7243	44.576	45.049	44.097	44.825	44.168	44.2438	44.2503	0.608	0.341	0.0849
9	5.0634	44.7508	44.265	44.101	44.802	44.698	44.597	44.291	44.1023	44.466	0.614	0.313	0.082
10	5.0846	44.7083	44.7531	44.227	44.358	44.544	44.857	44.25	44.4513	44.1819	0.628	0.338	0.085
11	5.0952	44.7561	44.3798	44.786	44.703	44.75	44.613	43.895	44.1589	44.1293	0.606	0.343	0.0867
12	5.1235	44.6338	44.3032	44.199	43.995	44.303	44.531	44.62	43.942	44.3713	0.657	0.354	0.0844
13	4.9714	44.6072	44.791	44.451	44.572	45.042	44.515	44.045	44.0835	44.1661	0.608	0.308	0.086
14	5.1129	44.7774	44.6191	44.395	44.67	44.956	43.945	44.127	44.3475	44.3029	0.638	0.333	0.0879
15	4.982	44.687	44.4181	44.772	44.028	44.733	43.929	44.1	44.0552	44.2029	0.659	0.369	0.0878
16	5.0669	44.7721	44.7435	44.94	44.506	44.458	44.466	44.633	44.1684	44.1503	0.611	0.432	0.0901
17	4.936	44.4796	44.6382	44.269	44.539	44.355	43.977	44.373	44.2815	44.2082	0.657	0.34	0.0875
18	5.1023	44.5168	44.2937	44.129	44.588	44.715	44.075	44.579	44.423	44.366	0.639	0.357	0.0847
19	4.9891	44.5806	44.4085	44.087	44.407	44.045	44.841	44.551	44.3853	44.3345	0.636	0.348	0.0848
20	4.9785	44.4583	44.5329	44.367	44.95	44.887	44.759	44.387	44.2155	44.4134	0.583	0.323	0.0908
21	5.0988	44.5221	44.3989	44.968	44.16	44.664	44.645	44.469	43.9514	44.1871	0.616	0.407	0.0898
22	5.1412	44.7933	44.6956	44.311	44.736	43.977	44.417	44.442	44.1401	44.3502	0.647	0.36	0.0858
23	5.1554	44.4902	44.2171	44.646	44.901	44.612	44.483	44.36	43.9986	44.3871	0.603	0.367	0.0856
24	5.1058	44.5062	44.2267	44.479	44.341	43.959	43.912	44.346	44.1212	44.224	0.683	0.363	0.0838
25	4.9678	44.7455	44.6669	44.828	44.868	44.492	43.994	44.278	44.0646	44.4029	0.622	0.36	0.0883
26	5.0138	44.4849	44.1884	44.493	44.127	44.337	44.548	44.059	44.1872	44.4555	0.655	0.339	0.0854
27	5.1377	44.6923	44.4659	44.814	44.325	44.2	44.254	43.908	43.9892	44.3976	0.661	0.359	0.0836
28	4.9572	44.7402	44.4851	44.143	44.473	44.166	44.336	43.963	43.9797	44.2976	0.669	0.29	0.0811
29	5.0386	44.5275	44.5425	44.339	44.456	44.905	43.961	44.086	43.9326	44.4239	0.647	0.326	0.0857
30	5.0422	44.7668	44.7626	44.52	44.177	44.819	44.629	44.401	44.0458	44.4344	0.621	0.364	0.0885
31	5.166	44.7189	44.3415	44.423	44.555	44.836	44.776	44.743	44.2721	44.2345	0.593	0.379	0.0869
32	5.12	44.5328	44.5616	44.059	44.967	44.423	44.401	43.936	44.2532	44.2713	0.647	0.288	0.0821
33	5.0598	44.5115	44.2362	44.632	44.687	44.63	44.906	44.196	44.489	44.2187	0.598	0.356	0.0865
34	4.9395	44.7827	44.3894	44.604	44.851	44.922	44.694	44.51	44.3004	44.2661	0.582	0.343	0.086
35	5.0563	44.6764	44.4564	44.856	44.72	44.269	44.89	44.715	44.1118	44.4502	0.595	0.421	0.0905
36	5.028	44.7242	44.3607	44.157	44.983	44.234	44.271	44.455	44.3381	44.124	0.634	0.345	0.0869
37	5.1589	44.5966	44.6095	44.744	44.391	44.561	43.896	44.702	44.0929	44.3608	0.637	0.424	0.0897
38	5.1306	44.7295	44.1788	44.213	44.209	44.389	44.32	44.031	44.3758	44.2555	0.676	0.322	0.0817
39	4.9608	44.6285	44.3702	44.842	44.621	44.681	44.792	44.018	43.8854	44.3555	0.599	0.341	0.084
40	5.1518	44.671	44.6478	44.255	44.226	44.475	44.222	44.209	43.9609	44.1345	0.667	0.339	0.0813
41	5.1342	44.5913	44.4468	44.325	44.259	44.973	44.955	43.99	44.074	44.3239	0.621	0.297	0.083
42	4.9962	44.7349	44.2841	44.534	44.374	44.08	44.043	44.496	44.357	44.445	0.659	0.389	0.0868
43	5.0492	44.7136	44.5234	44.562	44.605	44.372	44.743	43.881	44.4607	44.4607	0.628	0.329	0.0871
44	5.1271	44.469	44.5138	44.66	44.242	44.801	44.108	43.977	44.3664	44.2292	0.649	0.359	0.0845
45	4.9749	44.6551	44.5712	44.353	44.44	44.406	44.922	44.661	43.9043	44.1766	0.608	0.359	0.087
46	4.9997	44.7614	44.5521	44.073	44.275	44.647	44.01	44.729	44.1306	44.2608	0.651	0.367	0.0895
47	5.1483	44.5859	44.7818	44.898	44.835	44.767	44.678	44.182	44.2249	44.3292	0.591	0.379	0.0877
48	5.0032	44.5647	44.4277	44.241	45.016	44.028	44.189	44.606	43.9703	44.3766	0.637	0.357	0.0878
49	5.0174	44.5487	44.1693	44.171	44.522	44.853	44.45	44.319	44.0175	44.1608	0.621	0.314	0.0846
50	5.0775	44.4955	44.7339	44.381	44.489	44.131	44.808	44.223	43.9232	44.4081	0.637	0.343	0.0878
51	5.0528	44.5381	44.5904	44.758	44.917	44.286	44.157	44.072	43.876	44.1714	0.629	0.362	0.0852
52	5.1625	44.57	44.4946	44.73	44.094	44.062	44.711	44.414	44.4041	44.345	0.645	0.395	0.0821
53	4.9926	44.6179	44.6861	44.702	44.012	43.925	44.206	44.524	44.008	44.3082	0.668	0.403	0.0865
54	5.0457	44.453	44.5808	44.8	44.769	44.114	44.14	44.237	44.3192	44.4397	0.637	0.391	0.0874
55	4.9855	44.6125	44.198	44.548	44.934	44.578	44.026	43.922	44.291	44.3134	0.633	0.288	0.0883
56	5.1165	44.4743	44.5999	44.507	44.819	44.183	44.58	44.688	44.2344	44.1556	0.614	0.396	0.09
57	5.0882	44.6817	44.2745	44.912	44.308	45.025	44.368	44.332	44.3098	44.4292	0.611	0.388	0.0907
58	4.9502	44.6444	44.4372	44.185	44.144	44.99	44.434	44.114	44.4136	44.3397	0.638	0.3	0.0844
59	4.9431	44.5594	44.3128	44.618	44.292	44.784	44.303	44.77	44.0269	44.3923	0.618	0.378	0.0847
60	5.1094	44.6657	44.2554	44.409	44.753	43.942	44.939	44.155	44.0363	44.2398	0.632	0.33	0.0858
61	5.0315	44.6232	44.3511	44.59	44.045	44.509	44.287	44.674	44.4701	44.1398	0.64	0.452	0.0846
62	5.0209	44.5434	44.4755	44.87	44.884	45.008	44.059	44.565	44.2627	44.1977	0.598	0.374	0.0935
63	4.9537	44.554	44.6765	44.954	44.193	44.595	44.662	44.305	44.3947	44.3187	0.623	0.421	0.0906
64	5.0245	44.6498	44.7722	44.465	44.786	44.526	44.352	44.756	44.4796	44.3818	0.611	0.392	0.0912
65	5.0351	44.7029	44.6286	44.716	44.423	43.994	44.092	44.004	44.4418	44.1924	0.668	0.383	0.0859
66	5.0068	44.799	44.3224	44.688	43.979	44.32	44.874	44.264	44.2061	44.2766	0.651	0.338	0.0896
67	4.97	44.466	44.473	44.326	44.091	44.885	44.108	44.064	44.1	44.193	0.657	0.323	0.0841

.

Figure 3 reveals the relative effect and degree of a factor on an objective response based on a data set’s regression analysis. In this figure, the factors are listed in order of the largest effect to the smallest effect. The blue bars indicate positive effects on responses, while the red bars indicate negative effects. The ranking of effects is determined by ordering the scaled and normalized coefficients of a standard least-square second-order polynomial fit to the component’s data. This plot can be used to identify the factors with significant effects on the responses. As can be seen in the figure, the main factors influencing the C value are Y4, Y5, and Y6 with their negative degrees of 13.1%, 12.3%, and 12%, respectively. While quadratic components of Y5, r, and Y9 are the larger factors influencing on ${\overline{\epsilon }}_{S.D.}$ and $D_e^h$ with all negative degrees lower than 9%. Overall, no single factor can dominate any objective function. Response relationships between variables and objective functions appear to the complexity, which obviously increases the difficulty of optimum searching.

Figure 4 gives the distributions of initial sample data in the multi-objective space. As can be seen, the scattered red balls represent the spatial positions of each sample data in a coordinate system where three objective functions are chosen as the coordinate axes. The three groups of colored points are their vertical positions on different projection planes. In the projection plane of C and ${\overline{\epsilon }}_{S.D.}$, the projection points get an almost even distribution status over the whole region, which indicates that there is no obvious correlation between C and ${\overline{\epsilon }}_{S.D.}$. However, projection points in other projection planes are concentrated mostly on one half of the region. This implies an implicit correlation existed between some parameters. The Spearman rank correlation coefficient $\rho$ expressed as Equation (7) is often used to evaluate a correlation degree between two objective functions. The value of $\rho$ with 1 or −1 means a perfect monotonic positive correlation or negative correlation, respectively. However, no correlation exists when $\rho$ is equal to zero. As a result, there are certain correlations with coefficient of −0.4 between C and $D_e^h$, and 0.49 between $D_e^h$ and ${\overline{\epsilon }}_{S.D.}$. This means the forging performances can be optimized with improved deformation uniformity (lower ${\overline{\epsilon }}_{S.D.}$) which can decrease the probability of crack emergence (lower $D_e^h$) at the same time.

$$\rho =\frac{{\displaystyle \sum _{k=1}^Q(Z_k-\overline{Z})(S_k-\overline{S})}}{\sqrt{{\displaystyle \sum _{k=1}^Q{(Z_k-\overline{Z})}^2}}\sqrt{{\displaystyle \sum _{k=1}^Q{(S_k-\overline{S})}^2}}}$$

(7)

where Q is the number of sample, Z_k and S_k are the rank numbers of the kth objective value respectively, $\overline{Z}$ and $\overline{S}$ are the mean values of the Z_k and S_k respectively.

In addition, most of the samples locate the center of the parametric region. However, there are a few samples near the border, often the searching domain for the best solution, which poses a challenge in developing a reliable surrogate model for accurate predictions in this area.

5. Surrogate Models

For an optimization problem with ten variables, computational costs make it impractical to rely exclusively on high-fidelity simulations for the purpose of objective function evaluation. Therefore, an approximation model is essential for substituting an FE program to reduce the calculation costs during the optimization iteration. At present, response surface methodology, the Kriging model, and the radial basis function model are the main surrogate models applied in the metal forging area.

5.1. Response Surface Model (RSM)

A least-squares second-order polynomial fit is developed for analysis of the complex relationship between design variables and objective responses. Based on the initial sample data, a full second-order polynomial regression model including all two-way interactions is employed and expressed in Equation (8)

$$\tilde{y}={\beta }_0+{\displaystyle \sum _{i=1}^m{\beta }_i}{x}_i+{\displaystyle \sum _{i=1}^m{\beta }_{ii}}{x}_i^2+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1,j\ne i}^m{\beta }_{ij}{x}_i{x}_j}}$$

(8)

where $\tilde{y}$ is the objective response, ${\beta }_0$, ${\beta }_i$, ${\beta }_j$, and ${\beta }_{ij}$ are undetermined constant coefficients, $x_i$ and $x_j$ denote the independent design variables, and m is the number of design variables.

If $\delta$ is the model fitting error, the objective from sample data (y) can be expressed by:

$$y=\tilde{y}+\delta$$

(9)

Combined with Equations (8) and (9) can be transformed to matrix form as shown by:

$$\left[\begin{array}{c}{y}_1\\ y_2\\ \cdot \\ \cdot \\ \cdot \\ y_Q\end{array}\right]=\left[\begin{array}{ccccc}1& X_{11}& X_{12}& \cdot \cdot \cdot & X_{1k}\\ 1& X_{21}& X_{22}& \cdot \cdot \cdot & X_{2k}\\ \cdot & \cdot & \cdot & & \cdot \\ \cdot & \cdot & \cdot & & \cdot \\ \cdot & \cdot & \cdot & & \cdot \\ 1& X_{Q1}& X_{Q2}& \cdot \cdot \cdot & X_{Qk}\end{array}\right]\left[\begin{array}{c}{\beta }_0\\ {\beta }_1\\ \cdot \\ \cdot \\ \cdot \\ {\beta }_k\end{array}\right]+\left[\begin{array}{c}{\delta }_1\\ {\delta }_2\\ \cdot \\ \cdot \\ \cdot \\ {\delta }_Q\end{array}\right]$$

(10)

where Q is the number of samples and k is equal to $(m+1)\times (m+2)/2-1$(65 in this study). For simplicity, Equation (10) can be further expressed as:

$$\mathit{y}=\mathit{X}\mathit{\beta }+\mathit{\delta }$$

(11)

Because of Q > k + 1, the coefficient matrix of $\mathit{\beta }$ can be solved by the least squares regression which minimizes the residual sum of squares (RSS) between the sample data and approximated responses. Thus,

$$RSS={\displaystyle \sum _{i=1}^Q{(y_i-{\tilde{y}}_i)}^2}={\displaystyle \sum _{i=1}^Q{\delta }_i{}^2}$$

(12)

The deviations can be written as:

$$\mathit{\delta }=(\mathit{y}-\mathit{X}\mathit{\beta })$$

(13)

Then, the RSS becomes:

$$RSS={\mathit{\delta }}^T\mathit{\delta }=(\mathit{y}-{\mathit{X}\mathit{\beta })}^T(\mathit{y}-\mathit{X}\mathit{\beta })$$

(14)

The derivation of RSS with respect of $\mathit{\beta }$ can be obtained as follow:

$$\frac{\partial }{\partial \mathit{\beta }}(RSS)=-2{\mathit{X}}^T(\mathit{y}-\mathit{X}\mathit{\beta })=\mathbf{0}$$

(15)

Then, the coefficient matrix of $\mathit{\beta }$ can be obtained by solving the equation as follow:

$$\mathit{\beta }={({\mathit{X}}^T\mathit{X})}^{-1}{\mathit{X}}^T\mathit{y}$$

(16)

Coefficient tables of the regression model for all three objectives by an initial 67 sets of sample data are listed in

Table 2. Coefficients for response surface model (RSM) model based on initial sample data.

Term	C	${\overline{\mathit{\epsilon }}}_{\mathit{S}.\mathit{D}.}$	${\mathit{D}}_{\mathit{e}}^{\mathit{h}}$	Term	C	${\overline{\mathit{\epsilon }}}_{\mathit{S}.\mathit{D}.}$	${\mathit{D}}_{\mathit{e}}^{\mathit{h}}$
constant	755.803	−12243.136	−3298.667	r-Y4	0.010	0.024	−0.005
Y1	5.681	127.173	33.392	r-Y6	−0.011	0.000	0.022
r	−12.536	24.776	13.210	r-Y7	−0.037	0.016	0.032
Y2	−3.641	58.148	14.209	r-Y8	−0.046	0.023	0.005
Y3	−4.003	16.994	5.745	r-Y9	−0.036	0.102	−0.102
Y5	−1.424	20.805	5.616	Y2-Y3	−0.004	−0.009	−0.024
Y4	−1.796	6.834	2.890	Y2-Y5	−0.001	−0.002	0.008
Y6	−2.743	23.049	5.370	Y2-Y4	0.007	−0.097	−0.010
Y7	−2.848	30.270	8.629	Y2-Y6	−0.002	−0.055	−0.011
Y8	−2.933	44.311	11.196	Y2-Y7	−0.009	−0.226	−0.028
Y9	−2.502	220.859	60.002	Y2-Y8	−0.006	−0.082	−0.031
Y1^2	0.185	−1.578	−0.430	Y2-Y9	−0.003	0.038	0.014
r^2	0.142	−6.157	−1.473	Y3-Y5	0.005	0.023	−0.002
Y2^2	0.052	−0.503	−0.128	Y3-Y4	−0.006	0.032	0.002
Y3^2	0.026	−0.208	−0.051	Y3-Y6	0.014	−0.020	−0.013
Y5^2	0.009	−0.270	−0.066	Y3-Y7	0.016	0.010	−0.011
Y4^2	0.020	−0.113	−0.039	Y3-Y8	0.010	0.070	0.024
Y6^2	0.022	−0.183	−0.045	Y3-Y9	0.010	−0.015	−0.002
Y7^2	0.025	−0.206	−0.063	Y5-Y4	0.002	0.000	0.004
Y8^2	0.023	−0.572	−0.189	Y5-Y6	−0.001	0.033	0.000
Y9^2	0.014	−2.385	−0.669	Y5-Y7	0.006	−0.053	−0.014
Y1-r	−0.003	0.867	0.146	Y5-Y8	0.001	−0.047	0.005
Y1-Y2	−0.004	0.147	0.025	Y5-Y9	0.007	0.065	0.015
Y1-Y3	−0.003	−0.063	−0.004	Y4-Y6	−0.009	0.112	0.009
Y1-Y5	−0.005	0.067	−0.006	Y4-Y7	0.015	0.061	0.017
Y1-Y4	−0.011	0.144	0.010	Y4-Y8	0.004	−0.124	0.001
Y1-Y6	0.015	−0.010	0.012	Y4-Y9	−0.003	−0.061	−0.018
Y1-Y7	−0.015	−0.029	−0.012	Y6-Y7	0.001	−0.020	−0.007
Y1-Y8	0.006	0.180	0.106	Y6-Y8	−0.002	−0.085	−0.019
Y1-Y9	0.015	−0.226	−0.036	Y6-Y9	0.001	−0.110	−0.004
r-Y2	0.004	−0.137	−0.054	Y7-Y8	0.005	0.064	−0.004
r-Y3	−0.042	0.074	0.037	Y7-Y9	−0.001	−0.078	−0.014
r-Y5	0.010	−0.128	−0.044	Y8-Y9	0.006	0.162	0.041

, which contains all quadratic terms and all second-order cross terms.

5.2. Radial Basis Function (RBF) Model

An RBF surrogate model can be expressed as follows:

$$F(\mathit{X})={\displaystyle \sum _{p=1}^Q{w}_p}{\phi }_p(\mathit{X})+w_{Q+1}$$

(17)

where $F(\mathit{X})$ is the output response of any input vector (X), Q is the number of sample data, $w_p$ is the weight coefficient ascribed to the pth hidden output, $w_{Q+1}$ is the deviation compensation, and ${\phi }_p(\mathit{X})$ is the pth RBF expressed as follows:

$${\phi }_p(\mathit{X})={\Vert \mathit{X}-{\overline{\mathit{X}}}_p\Vert }^K$$

(18)

Here, $\Vert \mathit{X}-{\overline{\mathit{X}}}_p\Vert$ is the Euclidian distance given by $(\mathit{X}-{\overline{\mathit{X}}}_p{)}^T(\mathit{X}-{\overline{\mathit{X}}}_p)$. K is a shape function variable and 1.59 obtained by optimization. ${\overline{\mathit{X}}}_p$ is the center of the basis function ${\phi }_p(\mathit{X})$ and taken to be the corresponding given sample.

Clearly, the unknown coefficient $w_p$ and $w_{Q+1}$ need to be solved. The Q linear Equations can be obtained by introduction of the Q sample:

$${\displaystyle \sum _{p=1}^Q{w}_p}{\phi }_p(x_p)+w_{Q+1}=f_p,p=1,2,\dots ,Q$$

(19)

Introducing the notation:

$$\mathit{Z}=\left[\begin{array}{c}1\\ \cdot \\ \cdot \\ \cdot \\ 1\end{array}\right]\in {\mathit{R}}^Q, \mathit{G}=\left[\begin{array}{ccc}{\phi }_1(x_1)& \cdot \cdot \cdot & {\phi }_Q(x_1)\\ \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot \\ {\phi }_Q(x_1)& \cdot \cdot \cdot & {\phi }_Q(x_Q)\end{array}\right]\in {\mathit{R}}^{Q\times Q}, \mathit{H}=\left[\begin{array}{ll}\mathit{G}& \mathit{Z}\\ {\mathit{Z}}^T& 0\end{array}\right]\in {\mathit{R}}^{(Q+1)\times (Q+1)}$$

Equation (19) can be rewritten in the matrix form as:

$$\mathit{H}\cdot \mathit{w}=\mathit{F}$$

(20)

where w = (w₁,…,w_Q+1)^T, and F = (f₁,…,f_Q,0)^T. Then, the expansion coefficient matrix can be given by:

$$\mathit{w}={\mathit{H}}^{-1}\cdot \mathit{F}$$

(21)

Obviously, an added constant w_Q+1 constrains the sum of the weighted coefficients to zero, which can be described as:

$${\displaystyle \sum _{p=1}^Q{w}_p=0}$$

(22)

For the initialization of RBF model, least 2m + 1 sample data were required to be evaluated, where m is the number of design variables. However, as many samples as possible can contribute to enhancing the surrogate accuracy of an RBF model. An initial 67 sets of sample data were utilized for training the proposed RBF model. According to the computational method of the weighted coefficient, the results of respective expansion coefficients for three different objectives are shown in

Table 3. Regression expansion coefficients of radial basis function (RBF) model.

Serial Number of ω	C Value	${\overline{\mathit{\epsilon }}}_{\mathit{S}.\mathit{D}.}$	${\mathit{D}}_{\mathit{e}}^{\mathit{h}}$	Serial Number of ω	C Value	${\overline{\mathit{\epsilon }}}_{\mathit{S}.\mathit{D}.}$	${\mathit{D}}_{\mathit{e}}^{\mathit{h}}$
1	−0.0076	−0.0192	0.0013	35	−0.0070	−0.0520	−0.0029
2	−0.0002	−0.1173	−0.0385	36	−0.0019	−0.0103	0.0003
3	−0.0198	−0.2331	−0.0182	37	0.0066	0.0060	−0.0006
4	0.0067	0.0434	0.0010	38	−0.0170	0.0209	0.0046
5	0.0013	0.0413	0.0039	39	0.0116	−0.0331	0.0075
6	0.0090	−0.0444	0.0096	40	−0.0099	0.0289	0.0080
7	−0.0076	−0.0350	−0.0048	41	−0.0095	0.0197	0.0003
8	0.0005	−0.0009	0.0077	42	0.0059	0.0030	0.0050
9	0.0114	−0.0276	0.0080	43	0.0094	0.0180	−0.0016
10	0.0060	0.0113	−0.0027	44	0.0057	0.0181	0.0023
11	0.0192	−0.0243	−0.0008	45	0.0117	−0.0170	−0.0007
12	0.0013	0.0246	−0.0029	46	−0.0024	−0.0083	−0.0129
13	0.0121	0.0193	−0.0005	47	0.0044	0.0021	0.0075
14	−0.0060	0.0298	0.0013	48	−0.0075	−0.0230	−0.0039
15	−0.0012	−0.0261	−0.0074	49	0.0123	−0.0237	−0.0066
16	−2.5304	−0.0167	0.0024	50	−0.0007	0.0259	−0.0089
17	−0.0052	0.0435	−0.0028	51	0.0075	−0.0441	0.0020
18	−0.0050	0.0077	0.0073	52	0.0081	0.0822	0.0198
19	0.0123	0.0073	0.0016	53	0.0045	0.0307	0.0019
20	0.0102	−0.0011	−0.0126	54	0.0068	0.0009	0.0089
21	−0.0050	−0.0137	−0.0107	55	0.0005	0.0743	−0.0069
22	−0.0019	0.0562	0.0057	56	0.0026	−0.0072	−0.0055
23	0.0070	−0.0087	0.0086	57	0.0026	−0.0056	−0.0051
24	−0.0163	0.0592	0.0012	58	0.0088	0.0553	0.0046
25	0.0029	0.0333	0.0015	59	−0.0065	0.0184	0.0124
26	−0.0011	−0.0148	−0.0065	60	−0.0024	−0.0075	−0.0075
27	−0.0032	0.0156	0.0047	61	0.0124	−0.1480	0.0099
28	−0.0086	0.0195	0.0055	62	−0.0133	0.0681	−0.0041
29	−0.0022	−0.0066	−0.0008	63	−0.0182	−0.0901	−0.0078
30	0.0086	−0.0095	−0.0011	64	−0.0009	0.0339	0.0023
31	0.0116	0.0226	0.0035	65	0.0033	0.0060	0.0045
32	−0.0175	0.0379	0.0067	66	−0.0387	0.0941	−0.0156
33	0.0058	−0.0191	0.0037	67	−0.0058	−0.0073	0.0042
34	−0.0007	0.0164	0.0101	68	0.6311	0.3663	0.0885

. Note that the values with the serial number 68 are the added constants.

5.3. Kriging Model

The Kriging model postulates a combination of a ‘global’ linear regression model plus a ‘localized’ deviation:

$$y(x)=f{(x)}^T\beta +Z(x)$$

(23)

where $y(x)$ is the unknown function of interest, $f{(x)}^T\beta$, similar to RSM function, is a polynomial regression model with respect to x, which provides a ‘global’ model of the design space. $Z(x)$ is the realization of a stochastic process with mean zero, variance σ², and nonzero covariance, which represents a ‘localized’ deviation to the ‘global’ model. The covariance matrix of Z(x) that dictates the local deviations is:

$$Cov[Z(x),Z(v)]={\sigma }^2\mathit{B}([d(x,v)])$$

(24)

where B is the correlation matrix, and $d(x,v)$ is the correlation function between any two of the Q sample points x and v. B is a [Q × Q] symmetric, positive definite matrix with ones along the diagonal. The correlation function is user-specified and a Gaussian correlation function is adopted in this study:

$$d(x-v)={\displaystyle \prod }_{u=1}^m\exp (-{\theta }_u{\left|x_u-v_u\right|}^2)$$

(25)

where m denotes the dimension of input (the number of design variables), and ${\theta }_u$ are the unknown correlation parameters used to fit the model. Obviously, the Kriging model specification boils down to estimating the parameter $\beta$, the process variance σ², and the correlation parameter ${\theta }_u$. One can compare actual responses with model predictions for these experimental points used, then the unknown parameters and the Kriging model could be obtained by mathematical derivation. Once the best ${\theta }_u$ is confirmed, the predicted estimate $y(x_0)$ at an untried location $x_0$ can be given by:

$$y(x_0)=f{(x_0)}^T{\beta }_0+s^T(x_0){\mathit{B}}^{-1}(\mathit{Y}-\mathit{f}{\beta }_0)$$

(26)

In this study, $f{(x_0)}^T{\beta }_0$ is simplified to be a constant term $\widehat{\beta }$ when $f(x)$ is taken as a constant that is filled with ones. Y is the column vector of length Q that contains the sample values of the response. $s^T(x_0)$ is the vector of correlation values between the untried location x₀ and the sample data points:

$$s^T(x_0)={[d(x_0,x_1),d(x_0,x_2)\cdots d(x_0,x_Q)]}^T$$

(27)

The constant is estimated using the equation:

$$\widehat{\beta }={({\mathit{f}}^T{\mathit{B}}^{-1}\mathit{f})}^{-1}{\mathit{f}}^T{\mathit{B}}^{-1}\mathit{Y}$$

(28)

The estimate of the variance is:

$${\sigma }^2=\frac{{(\mathit{Y}-\mathit{f}\widehat{\beta })}^T{\mathit{B}}^{-1}(\mathit{Y}-\mathit{f}\widehat{\beta })}{Q}$$

(29)

The maximum likelihood estimate (MLE) for “best” ${\theta }_u$ are obtained by solving the following equation over ${\theta }_u>0$:

$$\max \underset{}{\varphi ({\theta }_u)}=-\frac{Q\ln ({\sigma }^2)+\ln \left|\mathit{B}\right|}{2}$$

(30)

where both $\widehat{\beta }$ and B are determined by ${\theta }_u$. The “best” Kriging model is found by solving the m-dimensional unconstrained nonlinear optimization problem given by Equation (30). An iterative optimization process is invoked to find the MLEs for searching the “best” ${\theta }_u$. The maximum iterations to fit the Kriging model are set to 3000.

Based on the initial 67 sets of sample data, the ${\theta }_u$, $\widehat{\beta }$ coefficients for normalized inputs are calculated and listed in

Table 4. ${\theta }_u$, $\widehat{\beta }$ coefficients and associated maximum likelihood estimates (MLEs) for three objectives.

Term	r	Y1	Y2	Y3	Y4	Y5	Y6	Y7	Y8	Y9	$\widehat{\mathit{\beta }}$	MLEs
C	0.0335	0.0587	0.0545	0.1046	0.2034	0.1562	0.1909	0.1241	0.0592	0.0114	0.1223	108.46
${\overline{\epsilon }}_{S.D.}$	0.0402	0.0189	0.0897	0.2201	0.135	0.1583	0.0641	0.2319	0.0406	0.0492	0.0846	64.48
$D_e^h$	0.0457	0.0828	0.1143	1.5682	0.2513	0.0039	0.0179	0.1272	0.0741	0.0095	0.2392	28.43

. The associated MLE values for three different objectives are also given at the same time.

6. Model Estimations and Comparisons

In this study, calculations of average error (ERR_ave) in Equation (31), maximum error (ERR_max) in Equation (32), root mean square error (ERR_rms) in Equation (33), and R-Square (R²) in Equation (34) have been employed to validate the model performances. These indicators have been normalized by the range of the actual values for each response, which allows the level of different responses with different magnitudes to be compared between different surrogate models.

$$ER{R}_{ave}={\displaystyle {\sum }_{i=1}^Q\left|y_i-{\widehat{y}}_i\right|}/Q\cdot (y_{max}-y_{min})$$

(31)

$$ER{R}_{max}=max\{\left|y_i-{\widehat{y}}_i\right|,i=1,2,\cdot \cdot \cdot Q\}/(y_{max}-y_{min})$$

(32)

$$ER{R}_{rms}=\sqrt{\frac{1}{Q}{\displaystyle {\sum }_{i=1}^Q{\left(y_i-{\widehat{y}}_i\right)}^2}}/(y_{max}-y_{min})$$

(33)

$$R^2={\displaystyle {\sum }_{i=1}^Q{({\widehat{y}}_i-\overline{y})}^2}/{\displaystyle {\sum }_{i=1}^Q{(y_i-\overline{y})}^2}$$

(34)

where ${\widehat{y}}_i$ is the predicted value by a surrogate model, $y_i$ is the corresponding true value from numerical simulation, $\overline{y}$ is the mean value of all true values, y_max and y_min are the maximum and minimum value among these true values, respectively. Lower values are desired for ERR_ave, ERR_max, and ERR_rms. No deviation exists between any true value and its corresponding prediction when the ERR_ave, ERR_max, and ERR_rms equal to zero. For the R-Square, a higher value is desired and the maximum value with one means no deviation between any true value and its corresponding prediction.

The fitting goodness to initial training sample data is very significant to evaluate the quality of a surrogate model. A reliable surrogate model should approximate close to these data.

Table 5. Fitting error analyses of different surrogate models based on the sample points. Abbreviations: average error (ERR_ave), maximum error (ERR_max), root mean square error (ERR_rms).

Surrogate Model	Objective Response	ERRave	ERRmax	ERRrms	R-Square
RSM	C	0.001	0.005	0.002	>0.999
	${\overline{\epsilon }}_{S.D.}$	<0.001	<0.001	<0.001	1
	$D_e^h$	0.003	0.009	0.003	>0.999
RBF	C	<0.001	<0.001	<0.001	1
	${\overline{\epsilon }}_{S.D.}$	<0.001	<0.001	<0.001	1
	$D_e^h$	<0.001	<0.001	<0.001	1
Kriging	C	0.005	0.014	0.006	>0.999
	${\overline{\epsilon }}_{S.D.}$	0.001	0.004	0.002	>0.999
	$D_e^h$	<0.001	<0.001	<0.001	1

lists the fitting performances of three different surrogate models at initial training sample points. As can be seen, all three surrogate models present excellent fitting accuracies. The RBF model fully interpolating all training samples represents a perfect fitting performance.

To further estimate the predictive capability of the developed surrogate models, an additional 10 sets of input data are created at random in the design domain for cross validation, as shown in

Table 6. Input data created at random for cross validation.

Number	r	Y1	Y2	Y3	Y4	Y5	Y6	Y7	Y8	Y9	C	${\overline{\mathit{\epsilon }}}_{\mathit{S}.\mathit{D}.}$	${\mathit{D}}_{\mathit{e}}^{\mathit{h}}$
1	5.022	44.67	44.695	44.597	44.992	44.344	44.306	44.308	44.121	44.23	0.617	0.366	0.0876
2	5.13	44.516	44.559	44.694	44.954	44.15	44.803	43.948	44.434	44.45	0.618	0.348	0.0861
3	4.981	44.459	44.472	44.635	44.522	44.972	44.274	44.561	44.324	44.146	0.609	0.369	0.0897
4	4.947	44.652	44.188	44.709	44.416	44.133	44.074	44.044	44.091	44.127	0.66	0.344	0.0847
5	5.079	44.535	44.187	44.281	44.549	44.971	44.669	44.707	43.879	44.344	0.594	0.358	0.0884
6	5.106	44.513	44.364	44.61	44.534	44.507	43.989	44.177	44.375	44.464	0.648	0.365	0.0862
7	5.117	44.493	44.649	44.382	44.314	44.983	44.427	44.054	44.339	44.369	0.633	0.329	0.0862
8	5.147	44.592	44.702	44.658	44.393	44.652	44.463	43.927	44.334	44.236	0.636	0.35	0.0845
9	5.000	44.538	44.285	44.105	45.02	44.163	44.538	43.954	44.083	44.296	0.64	0.281	0.0824
10	5.145	44.527	44.528	44.664	44.417	44.237	43.924	44.198	44.212	44.276	0.664	0.381	0.0845

. Response fit plots (Figure 5) show true values versus predicted values for each response of all developed approximation models. The blue horizontal line gives the mean response value predicted by surrogate models. The diagonal line in the plot represents a perfect fit. From these figures, all three models provide good fit accuracies to the response of the C value because the checkpoints are very close to the diagonal line. However, the approximation qualities of other responses differ from different models. The predictions of the response $D_e^h$ are relatively poor-quality in all three models.

Specific fitting errors of different surrogate models based on the additional points have been listed in

Table 7. Fitting error analyses of different surrogate models based on the additional points.

Surrogate Model	Objective Response	ERRave	ERRmax	ERRrms	R-Square
RSM	C	0.039	0.075	0.044	0.979
	${\overline{\epsilon }}_{S.D.}$	0.549	1.184	0.668	0
	$D_e^h$	2.009	4.384	2.441	0
RBF	C	0.018	0.039	0.021	0.995
	${\overline{\epsilon }}_{S.D.}$	0.065	0.104	0.071	0.928
	$D_e^h$	0.123	0.444	0.173	0.611
Kriging	C	0.058	0.123	0.071	0.945
	${\overline{\epsilon }}_{S.D.}$	0.071	0.119	0.082	0.905
	$D_e^h$	0.161	0.561	0.226	0.341

. As can be seen, the R-Square values in response to C are mostly close to 1.0 in all models, which confirms that good fitting qualities have been achieved by all models. It further proves the objective of C can be characterized easily by a second-order polynomial. However, R-Square values of response ${\overline{\epsilon }}_{S.D.}$ and $D_e^h$ are zero by the RSM model. Especially for response $D_e^h$, both ERR_ave and ERR_rms are larger than one. That means the average predicted errors have far exceeded the range of the actual values, and the predictions turn out to be a complete failure. The objective function of $D_e^h$ takes on the characteristics of complex nonlinearity.

However, the RBF and Kriging models possess relatively good fitting precisions in response to ${\overline{\epsilon }}_{S.D.}$ because of both R-Square values larger than 0.9. By comparison, predictive qualities of $D_e^h$ decline from both the Kriging and the RBF models. In Table 7, the fitting errors of R-Square value are 0.611 and 0.341, respectively. As a whole, the RBF model is better than the Kriging model in predictions of all the objectives. In order to increase the fitting accuracy, both the initial training sample and additional points for cross validation are employed to retrain the RBF model for the following optimization process.

7. The Shape Optimization of an Aerofoil Preform Tool

7.1. Particle Swarm Optimization Algorithm

Particle swarm optimization (PSO) [25,26] is a population-based global search procedure where individuals (called particles) continuously change positions within the search area. The state of a particle is the value for all design variables in an optimization problem and its velocity in the design space, and each move produces a new generation. In each iteration, the particle is moved using two values: its own best position and the global best position by all particles among all the generations. Compared with GA-based algorithms, there are no complex genetic procedures required, which significantly improves the efficiency of the optimum searching by using a PSO algorithm. The detailed introduction for a PSO algorithm can be referred to the literature [27].

Table 8. Parameters used for the particle swarm optimization (PSO) algorithm in this study.

Parameter Name	Value	Description
Maximum iterations	100	Maximum number of iterations during optimization
Number of particles	100	The size of the population
Inertia	0.9	The inertia of each particle in the swarm
Global increment	0.9	The maximum increment to the particle position based on the global best position
Particle increment	0.9	The maximum increment to the particle position based on the particle best position
Maximum velocity	0.1	The maximum value of the particle velocity

lists the parameters of the PSO algorithm used in this study.

7.2. Design of Fitness Function

One important step for multi-objective optimization is to define the fitness function which decides the searching direction for an optimized solution. In this study, a zero-mean normalized process is employed for the conversion of objective function value. Each value will be converted into the number of the standard deviation away from the mean. The mean and the standard deviation can be estimated from the samples. This normalized approach is very suited in cases where there are unknowns about the value range of objective function in advance. As a result, a fitness function is suggested based on a weighted treatment of the normalized multi-objective function and a penalization of the solution according to its constraint violation, which is defined as Equation (35). A PSO algorithm could help the search towards more promising regions in the design space, and to seek to decrease the fitness value of $F_i$ continually.

$$F_i=w_1\cdot F_{{\overline{\epsilon }}_{{}_{S.D}}}{}_{,i}+w_2\cdot F_{D_e^h,i}+F_{p,i}$$

(35)

where $F_{{\overline{\epsilon }}_{{}_{S.D}}}{}_{,i}$ is the normalized value of ${\overline{\epsilon }}_{S.D.}$ of individual i, $F_{D_e^h,i}$ is the normalized value of $D_e^h$ of individual i, $F_{p,i}$ is the penalty term of individual i, and $w_1$ and $w_2$ are the weighting factors and assigned to 1 in this study. $F_{{\overline{\epsilon }}_{{}_{S.D}}}{}_{,i}$, $F_{D_e^h,i}$, and $F_{p,i}$ are respectively defined as:

$$F_{{\overline{\epsilon }}_{{}_{S.D}}}{}_{,i}=({\overline{\epsilon }}_{{}_{S.D,i}}-{\mu }_{\overline{\epsilon }})/{\sigma }_{\overline{\epsilon }}$$

(36)

$$F_{D_e^h,i}=(D_{e,i}^h-{\mu }_D)/{\sigma }_D$$

(37)

$$F_{p,i}=\{\begin{array}{ll}0,& if a_i\le 0.001\\ 1+1000\times a_i,& if a_i>0.001\end{array}$$

(38)

where ${\overline{\epsilon }}_{{}_{S.D,i}}$ and $D_{e,i}^h$ are the standard deviation of effective strain and the average of 1% elements with highest damage value of individual i during the evolutionary process; ${\mu }_{\overline{\epsilon }}$, ${\sigma }_{\overline{\epsilon }}$, ${\mu }_D$, and ${\sigma }_D$ are the mean values and the standard deviation of respective ${\overline{\epsilon }}_{S.D.}$ and $D_e^h$ column in the sampling data, which are 0.358, 0.0361, 0.0866, and 0.00316 by calculation; and $a_i$ is the die-filling status of individual i, as defined in Equation (1).

7.3. Optimization Result

Figure 6 shows the distributions of solutions for a total of 10,000 individuals. It can be seen that almost all solutions cluster together near the diagonal line, which further proves the positive linear correlation between objective values ${\overline{\epsilon }}_{S.D.}$ and $D_e^h$. The distinctive green point represents the optimum solution in all individuals according to the minimum of fitness function.

Figure 7 shows the fitness value distributions in the individual evolutionary process. It can be seen that most fitness values are from −2 to −4 and close to the optimum value, which reveals a high searching efficiency. After about 3,000 evolutions, the minimum of fitness value declines gently and tends to be constant. The searching process expresses good stability and convergence, and finally presents an optimum fitness value of −5.388 marked by the green point. Compared with the minimum fitness value of −3.641 in sample data (with serial number of 28 in Table 1), the optimal result improves considerably. The objective values and the associated design variables corresponding to the optimum solution predicted by the RBF model are listed as follow:

$$\{\begin{array}{l}\mathrm{Variables}:Y1[44.799],Y2[44.169],Y3[44.059],Y4[45.049],Y5[45.042]\\ Y6[44.955],Y7[43.881],Y8[43.798],Y9[44.466],r[4.936]\\ \mathrm{Objectives}:C=0.587;{\overline{\epsilon }}_{S.D.}=0.238;D_e^h=0.0797\end{array}$$

According to the predicted solution, a new upper die profile of preform tool can be redesigned. Then, FE simulations with the optimal billet and preform tool for the pre-forging process and the final forging process have been implemented to validate the optimization result as shown in Figure 8 and Figure 9. It can be seen that the preform shape is relatively simple and nearly symmetrical. The corresponding C value is 0.577 which is quite approximate to the prediction (0.587), which no doubt is in favor of the preparation of the preform. As a whole, the deformation is relatively large in the pre-forging stage for pursuing a better pre-distribution of material.

However, in the final forging process, the local high strain and overall deformation degree have both declined comparatively. This will contribute to enhancing forming qualities and reducing forging defects. In Figure 9, the high damage values totally concentrate on the two sides near the flash area where high effective strains exist. The values of ${\overline{\epsilon }}_{S.D.}$ and $D_e^h$ are 0.249 and 0.0827 from the result of FE simulations which also approximate the predictions (0.238 and 0.0797). In addition, a sufficient die-filling status and the uniform simplified flash have been achieved at the same time. Note that the optimized billet size (r = 4.936) is near the lower limit in the range of r value, which indicates that the billet with reasonably reduced material contributes to the improvement of material flow and forming properties.

8. Conclusions

In this paper, the RSM, RBF, and Kriging surrogate models were developed to substitute the FE simulation process for predicting the objective responses efficiently. A comparison of the three surrogate models were implemented for the evaluation of fitting accuracy based on a shape design and optimization of an aerofoil forging preform tool. According to a parametric study, the complex correlations between design variables and objective functions were clarified to a certain degree. Quantitative analyses in detail manifest that although achieving the excellent fitting qualities at the training points in all surrogate models, the predictive performance differed in different models when responding to the general points in the variable space. The response of C can be well characterized by a simple second-order polynomial, and all suggested surrogate models provided better fitting effects. To the moderately complex objective function of ${\overline{\epsilon }}_{S.D.}$, the RBF and Kriging models still presented good fitting performance compared to the RSM model which totally lost its predictive ability. As for the response of $D_e^h$ with high nonlinearity, more training samples are necessary for improving the predictive precision and reliability to all existing models. In this case, the RBF model was slightly superior to the Kriging model in prediction of the $D_e^h$.

An optimum design of preform tool was determined by the PSO algorithm which assisted the proposed RBF model, and which has the ability to significantly improve forging qualities. The optimized preform tool was validated by the FE simulation. The forging performances derived were proved to be very close to the predictions from the optimization scheme. The result also shows that the suggested RBF model has a preferable surrogate accuracy, and could be efficiently applied to handle complex engineering optimizations.