Carbon fiber reinforced plastics (CFRPs) have attracted a growing attention from transportation industry for their high performance-to-weight ratios, excellent dimensional stability, and good corrosion and fatigue resistance [1; 2] . Conventionally, CFRP parts are regarded to be expensive and their application is mostly limited to small-volume production such as manufacturing of airplanes. Recently, however, demand for light-weight vehicles has increased drastically due to concerns over excessive fuel consumption and carbon emission [3; 4] . As a result, various automatic approaches for mass-production of CFRP parts at low cost, such as resin transfer molding (RTM) [5] [6] [7] , pultrusion [8; 9] and thermoforming [10] [11] [12] , have been designed and developed. Among these approaches, thermoforming can provide the best combination of automation, production rates, part geometrical complexity and cost efficiency, therefore it is the most promising method to manufacture CFRP parts in large volume and at low cost.
Thermoforming consists of a highly-automated 2-step process chain including preforming and curing, and the flowchart of thermoforming is depicted in Fig. 1 . The raw materials are woven carbon fiber prepregs, which are woven carbon fabrics impregnated with polymer resins. In preforming, woven prepregs are heated over the melting temperature of the resin to soften the material and then these hot prepregs are formed by a press. In the following resin solidification step, the preformed part is secured in a mold and the resin is solidified to fix the shape of the part either by temperature decrease for thermoplastic materials or by high pressure and temperature for thermoset materials. Finally, after some essential trimming and polishing work, CFRP parts with target geometries can be obtained.
Carbon fiber reinforced plastics (CFRPs) have attracted a growing attention from transportation industry for their high performance-to-weight ratios, excellent dimensional stability, and good corrosion and fatigue resistance [1; 2] . Conventionally, CFRP parts are regarded to be expensive and their application is mostly limited to small-volume production such as manufacturing of airplanes. Recently, however, demand for light-weight vehicles has increased drastically due to concerns over excessive fuel consumption and carbon emission [3; 4] . As a result, various automatic approaches for mass-production of CFRP parts at low cost, such as resin transfer molding (RTM) [5] [6] [7] , pultrusion [8; 9] and thermoforming [10] [11] [12] , have been designed and developed. Among these approaches, thermoforming can provide the best combination of automation, production rates, part geometrical complexity and cost efficiency, therefore it is the most promising method to manufacture CFRP parts in large volume and at low cost.
Thermoforming consists of a highly-automated 2-step process chain including preforming and curing, and the flowchart of thermoforming is depicted in Fig. 1 . The raw materials are woven carbon fiber prepregs, which are woven carbon fabrics impregnated with polymer resins. In preforming, woven prepregs are heated over the melting temperature of the resin to soften the material and then these hot prepregs are formed by a press. In the following resin solidification step, the preformed part is secured in a mold and the resin is solidified to fix the shape of the part either by temperature decrease for thermoplastic materials or by high pressure and temperature for thermoset materials. Finally, after some essential trimming and polishing work, CFRP parts with target geometries can be obtained. 
Carbon fiber reinforced plastics (CFRPs) have attracted a growing attention from transportation industry for their high performance-to-weight ratios, excellent dimensional stability, and good corrosion and fatigue resistance [1; 2] . Conventionally, CFRP parts are regarded to be expensive and their application is mostly limited to small-volume production such as manufacturing of airplanes. Recently, however, demand for light-weight vehicles has increased drastically due to concerns over excessive fuel consumption and carbon emission [3; 4] . As a result, various automatic approaches for mass-production of CFRP parts at low cost, such as resin transfer molding (RTM) [5] [6] [7] , pultrusion [8; 9] and thermoforming [10] [11] [12] , have been designed and developed. Among these approaches, thermoforming can provide the best combination of automation, production rates, part geometrical complexity and cost efficiency, therefore it is the most promising method to manufacture CFRP parts in large volume and at low cost.
Thermoforming consists of a highly-automated 2-step process chain including preforming and curing, and the flowchart of thermoforming is depicted in Fig. 1 . The raw materials are woven carbon fiber prepregs, which are woven carbon fabrics impregnated with polymer resins. In preforming, woven prepregs are heated over the melting temperature of the resin to soften the material and then these hot prepregs are formed by a press. In the following resin solidification step, the preformed part is secured in a mold and the resin is solidified to fix the shape of the part either by temperature decrease for thermoplastic materials or by high pressure and temperature for thermoset materials. Finally, after some essential trimming and polishing work, CFRP parts with target geometries can be obtained. 
Carbon fiber reinforced plastics (CFRPs) have attracted a growing attention from transportation industry for their high performance-to-weight ratios, excellent dimensional stability, and good corrosion and fatigue resistance [1; 2] . Conventionally, CFRP parts are regarded to be expensive and their application is mostly limited to small-volume production such as manufacturing of airplanes. Recently, however, demand for light-weight vehicles has increased drastically due to concerns over excessive fuel consumption and carbon emission [3; 4] . As a result, various automatic approaches for mass-production of CFRP parts at low cost, such as resin transfer molding (RTM) [5] [6] [7] , pultrusion [8; 9] and thermoforming [10] [11] [12] , have been designed and developed. Among these approaches, thermoforming can provide the best combination of automation, production rates, part geometrical complexity and cost efficiency, therefore it is the most promising method to manufacture CFRP parts in large volume and at low cost.
Thermoforming consists of a highly-automated 2-step process chain including preforming and curing, and the flowchart of thermoforming is depicted in Fig. 1 . The raw materials are woven carbon fiber prepregs, which are woven carbon fabrics impregnated with polymer resins. In preforming, woven prepregs are heated over the melting temperature of the resin to soften the material and then these hot prepregs are formed by a press. In the following resin solidification step, the preformed part is secured in a mold and the resin is solidified to fix the shape of the part either by temperature decrease for thermoplastic materials or by high pressure and temperature for thermoset materials. Finally, after some essential trimming and polishing work, CFRP parts with target geometries can be obtained.
48th SME North American Manufacturing Research Conference, NAMRC 48 (Cancelled due to COVID-19) Figure 1 shows that thermoforming involves numerous process parameters, all of which need to be designed to efficiently produce parts with optimal performance. One important parameter that must be considered is the initial blank geometry. The initial blank geometry determines the trimming step, during which excessive prepreg along the edges of the part, as shown by the example in Fig. 2 , will be cut away. Too much trimming means additional production time, waste of expensive raw material, and wear of the trimming tools, constituting a significant problem in mass-production. Blank geometry design is important to thermoforming as well as all other forming processes. It also has been researched extensively for conventional metal forming because this process has already been popular in industry [13; 14] Various design optimization methods, such as the slip-line [15; 16] , characteristic of plane stress [17; 18] , and geometric mapping [19] methods and inverse FEM modeling [20] [21] [22] , have been developed successfully.
However, it is highly unlikely that established techniques for metal forming can be directly transferred to prepreg blank shape design in thermoforming because composite prepregs have higher anisotropy than metals and thermoforming is a complex process involving multidisciplinary considerations in thermal, mechanical and physical aspects [23] . Therefore, new approach need to be developed to tackle this challenge.
Thermoforming of CFRPs is a recently-developed manufacturing technique and its application is still mostly limited to lab-scale or small-volume production which are not cost-sensitive, so the existing blank geometry design methods are primitive and relies on rough estimation by pure experiments [24; 25] . General industrial practice is to mark grids on the surface of the blank before preforming. Deformation of these grids helps to understand how the material deforms and what the optimal blank geometry should be. Density of the grids is usually coarse due to limitation of ink, surface properties and optical resolution. As a result, this method only works well for (1) simple part geometries when the draw-in distances of the prepregs are uniform, and (2) small batch production when the total trimming cost is controlled by relatively low production volume. For more complex and performance-optimized geometries that are essential for commercialized mass-production in industry, the grid method can hardly achieve optimal blank geometry. Alternatively, FEA models of thermoforming, once correctly developed, can capture deformation throughout the prepreg blanks in order to precisely identify regions of the blanks to be preformed or trimmed in the final part. Hence, compared to pure experimental methods, the FEA simulation method has great potential to design prepreg blank geometry with little excessive material to be trimmed after thermoforming.
Although using FEA simulation to design prepreg blank geometry is very promising, it is seldom utilized in practice nowadays for the following reasons: (1) previous FEA models of CFRP thermoforming, such as the ones in referecens [3; [26] [27] [28] , lack required level of prediction accuracy and/or application convenience; (2) not until recently has the production of CFRP parts been in small volumes, so the material waste and tool wear cost associated with trimming was not such a major problem; and (3) most of the CFRP parts produced currently have simple geometries, thus rough estimation from experiments is sufficient.
To meet demands from process design for CFRP thermoforming, a cutting-edge FEA modeling method for preforming has been developed based on the non-orthogonal material frame and realized as user-defined material subroutine in commercial software such as LS-DYNA [29] and Abaqus [30] . Comparison of the simulation results with industrial benchmark experiments, as documented in references [29; 30] , indicates that this FEA modeling method successfully captures the prepreg deformation, including draw-in distances, wrinkling appearance and variation of angles between warp and weft yarns, caused by preforming. Since most material deformation in thermoforming processes is induced by the preforming step while the curing step has negligible influence on the thermoformed part geometry, this preforming modeling method alone can provide accurate digital representation of thermoforming, and in this paper, the modeling method is integrated with automatic mesh adjustment to form an innovative approach as foundation for initial blank geometry optimization in order to minimize prepreg trimming. To illustrate this integrated approach for blank geometry design, the following paper is organized as such: Section 2 briefly introduces the FEA modeling method and automatic mesh adjustment for blank optimization; Section 3 explains preliminary results and effectiveness of the integrated approach for single-layer and multiple-layer thermoforming/preforming when trial-and-error optimization is applied; and Section 4 provides conclusion and future directions. Due to experimental limitations, the designed blank geometries only include preliminary FEA simulation without physical validation, but the results are relatively trustworthy since the FEA modeling method, as mentioned above, has already been validated by benchmark tests.
In order to simulate the prepreg anisotropy dominated by fiber directions, the preforming FEA model describes the material constitutive law in the non-orthogonal material frame formed by warp and weft yarns during preforming, as shown in Fig. 3 . Material properties input to the model can be obtained via either pure experimental [31] or numerical characterization [30] . Both inputs generate high accuracy in part geometry prediction [29; 30] , and the pure experimental input is selected for the blank design application because of its simplicity. Woven CFRP prepregs are highly anisotropic, so initial fiber orientations and stacking configurations, in addition to target part shape, need to be considered in initial blank geometry design for preforming. Mathematically, the design formulation can be constructed as:
where is the blank geometry in simulation, (•) is the preformed part shape predicted by simulation using , is the target part shape, is the stacking sequence and * is the optimal blank geometry. In the reported integrated approach using FEA simulation, initial and preformed geometries are defined by elements. The optimal blank geometry is achieved when boundary curve of target part goes through all boundary elements in the preformed part predicted by simulation. Hence, the objective function [•; , ] can be written more specifically as:
where includes all boundary elements and contains all nodes in each boundary element. (•; ) represents node coordinates in the preformed part, which are controlled by and . (•; ) is the calculation of distance: if one node is on the outside (inside) of target geometry, is positive (negative). The reported integrated approach will serve as foundation for preforming blank design from two aspects: (1) it provides cost-effective numerical models to represent processes; and (2) it can calculate objective function expressed by equation (2). To form a whole design procedure and illustrate effectiveness of the new approach, basic trial-and-error optimization is applied in current work for preliminary analysis, and the flowchart of the blank geometry design loop is plotted in Fig.  4 . Mesh adjustment operation is automated via Python scripts and MATLAB ® codes for user convenience. The design iteration starts from a predefined blank geometry used in Fig 2, which was determined by experience and requires trimming after preforming. The predefined geometry is then input to FEA preforming simulation. After simulation, geometry of the preformed part is compared to the target one. If the simulation result does not meet a criterion Δ for blank shape modification, the optimal blank geometry is regarded as the geometry used in the preforming simulation. If the result meets Δ, on the other hand, the flowchart goes to the mesh adjustment algorithm to modify the elements that discretized the prepreg blanks and maps these elements back to the 2D prepreg blanks for the next iteration of the preforming simulation. One advantage of using FEM models to design blank geometry is that the prepreg blanks are discretized by elements at the mm length scale, so that the adjustment of the blank geometry can be more refined compared to experimental methods. In current practice, the mesh of prepreg blanks consists of S4R elements, and the criterion Δ for mesh adjustment includes two scenarios with trial-and-error optimization: (1) one element has all four nodes outside the target geometry, which means the blank is larger than the ideal one and additional trimming is necessary, or (2) one element that was on the boundaries of the target geometry in the last design iteration has all four nodes inside the target geometry in this iteration, which means the designed blank is actually smaller than the ideal one and too much trimming has been done. If either scenario happens and Δ is met, the design will go to the mesh adjustment algorithm and then to next iteration with the new blank geometry. This criterion ensures that the optimal blanks require as little trimming as possible while the remaining material can fully cover the target geometry, as detailly demonstrated in the following.
For the sake of element rebirth, the predefined blank geometry is intentionally designed to be large so that it can fully cover the target part geometry after preforming. For mesh adjustment, deletion of excessive elements to generate a new blank mesh is straightforward, but the prepregs have complex mechanical properties, so in the next iteration, as shown in Fig.  5 , the modified blanks with deleted elements and trimmed mesh may result in smaller parts than the target. To solve such possible manufacturing defects, a systematic mesh adjustment technique with both element deletion and element addition is developed and implemented into the blank geometry design method. Its flowchart is shown in Fig. 6 . For each design iteration, both scenarios of Δ will be checked and addressed to ensure that the blank geometry after design will be optimal. During the numerical design procedure, the mesh adjustment algorithm will be initiated when Δ is met. This algorithm first checks all the elements in the preformed prepregs in sequence. In the current work, all elements in the prepregs are S4R. After each preforming simulation, if all four nodes of an element are outside the target geometry, this element is in the excessive region, so it will be deleted with corresponding nodes in the next design iteration. If all four nodes of an element are inside the target geometry, this element is in the target part region and kept for the next iteration. The above mesh adjustment addresses scenario (1) of Δ, i.e. when the blank is too large. Meanwhile, all other elements are partly inside the target region, so they are grouped into Bi in which i means the iteration number and starts from 1. The retained elements and the corresponding nodes make up the new mesh of the prepreg blanks for preforming simulation in the next iteration. Then if the mesh adjustment algorithm is initiated again and i is larger than 1, a sequential check for elements to be either deleted or retained will be performed one more time. Moreover, the elements in group Bi-1, which are the boundary elements in the last simulation iteration, will be checked. Any element in Bi-1 moving into the target region indicates the appearance of gaps as illustrated by Fig. 5 and these elements are denoted as B'i-1. To ensure preforming quality after blank design, the neighboring elements of B'i-1 that have been deleted in iteration i-1 will be restored and added to Bi. This adjustment addresses scenario (2) of Δ, that is when the blank is too small, by adding more material to the positions at the edges of the blanks where gaps appear between the simulated preformed part and the target part. In this way the mesh adjustment algorithm aids the FEA-modeling-based design to achieve the blank geometry that requires as little trimming as possible without gaps.
To test its effectiveness, the integrated approach is applied to preforming simulation with one and two prepreg layers using trial-and-error optimization. Simulation setup, as shown in Fig.  7 , is the same for the one-and two-layer preforming. The double-dome benchmark tool geometry [32] is used since it contains all geometrical features expected in CFRP parts for high-volume automobile industry. Tools, including punch, binder and die, are modeled as rigid bodies by shell elements. During preforming, positions of the binder and die are fixed while movement of the punch is controlled by displacement. The prepreg layers are discreted using S4R with the size of 4 mm by 4 mm with five through-thickness integration points. For selected prepreg materials, the appropirate preforming temperature is 60 °C because under this condition, resin melts, prerpegs have high formibility in shear, and no wrinkle appears with displacement-controlled binder. Hence, experimentally characterized material properties at 60 °C are input into the non-orthogonal material law to eliminate wrinkling affected by other process parameters instead of initial blank geometry. As for surface properties, the prepreg-tool interaction is simulated by Coulomb friction coefficient of 0.2 while the prepregprepreg interaction in the two-layer simulation is simulation by Coulomb friction coefficient of 1.6 from the measurement recorded in [33] . The FEA simulation is performed by the ABAQUS dynamic explicit solver, and the design results are elaborated by the following 2 subsections. Fig. 7 Setup for double-dome preforming simulation.
In one-layer initial blank geometry design, one prepreg sheet with initial fiber orientation of 45 º is preformed. Definition of the fiber orientation is illustrated in Fig. 8 (a) and the selected 45 º angle is a popular choice in industry. The predefined blank geometry determined by an industrial lab is also shown in Fig. 8 (a) . This predefined geometry has length of 688 mm and width of 454 mm. The experimentally preformed part shape using the predefined blank geometry is illustrated in Fig. 8 (b) , while the target part geometry is shown in Fig. 8 (c) . The comparison between Fig. 8 (b) and (c) indicates that trimming is still essential after the experimental blank design. Alternatively, in the design procedure based on the newly developed integrated approach, the predefined blank geometry is first input to the preforming simulation. As has been validated in [29] , the simulation result shown in Fig. 9 (a) agrees very well with the experimental one in Fig. 8 (b) . Then using the newly developed modeling-based design method, the optimal blank geometry shown in Fig. 9 (b) is obtained. As demonstrated in Fig. 9 (c), this optimal blank geometry leads to the preforming result that is very close to the target geometry shown in Fig. 8 (c) apart from some jagged edges. As for raw material cost evaluation, the optimal blank has length of 588 mm and width of 400 mm, leading to surface area of 2010 cm 2 , which is over 30% smaller than the 2887 cm 2 of the predefined blank. 
After blank design of one-layer preforming, same integrated approach and trial-and-error optimization is applied to twolayer preforming. Both prepreg layers have the same predefined geometry shown in Fig. 8 (a) , but the top layer has the fiber orientation of 0 ° while the bottom layer has orientation of 45 °. Material shrinkage during curing suggests that this asymmetric stacking configuration may lead to final warpage and is seldomly used in practice. However, this setup is suitable for pure preforming analysis and it can consider interaction between different prepreg layers with lowest computational cost. The preforming simulation result with the predefined geometry is shown in Fig. 10 (a) . Compared to the desired shape shown in Fig. 8 (c) , trimming of excessive materials is essential. Then by using the new modeling-based design method, the optimal initial blank geometries for both prepreg layers can be obtained, as demonstrated in Fig. 10 (b) . Preforming simulation result using these optimal blank shapes is shown in Fig. 10 (c) . Similar to the one-layer case, large amount of trimming is no longer necessary to obtain the desired part geometry except for the jagged edge. As for material cost analysis, surface area of the predefined blank is 2887 cm 2 . For the optimal blanks, the areas of 0 ° and 45 ° blanks are 2019 cm 2 and 2018 cm 2 , respectively, both of which are over 30% smaller than the predefined blank. For both one-layer and two-layer preforming processes, the optimal results have jagged edges, because the modeling-based design method modifies the blank geometry based on the mesh of the predefined blank. This phenomenon can be improved by planning the real cutting line of the prepreg blanks to be the smallest smooth envelop for the optimal blank geometry. Then trimming of the part made by preforming can be kept at minimum level.
One interesting fact noticed in this design procedure with the new integrated approach is that the optimal blank geometry, such as the ones in Fig. 9 (b) and Fig. 10 (b) , can be achieved after only one iteration in the double-dome benchmark setup, while four to six iterations are usually required for design in metal forming [34] . The reason is that compared to metals, woven prepregs during preforming have very high anisotropy dominated by stiff carbon fibers. For example, the 2x2 twill prepreg selected for this work has tensile moduli along carbon fiber yarns at 10 GPa level while shear modulus of woven fabric at 0.1 MPa level. As a result, the dominant deformation mode of the prepregs is pure shear, and materials in the trimmed regions have negligible influence on strain and yarn orientation distributions of the materials in the target geometry region. As a clear demonstration, at the end of preforming, distributions of angle between warp and weft yarns throughout the double-dome parts predicted by the preforming simulation with predefined and optimal blank geometries are shown in Fig. 11 . The comparison implies that before and after deleting excessive prepregs on the outside, difference of yarn angle distributions is insignificant, supporting the idea that during preforming, deformation of prepregs within the target geometry is almost independent from constraint of the excessive prepregs. This phenomenon also means that for most of the geometries covered by the double-dome benchmark, the FEA-modeling-based blank design can be quickly completed after just one round of simulation without any further back and forth mesh adjustment, given arbitrary starting blank geometry that is large enough. To further verify the statement that one design iteration can lead to optimial blank geometry for double-dome benchmark, two other starting blank geometries that are easy to be determined are tested. As shown in Fig. 12 , one geometry is rectangular that envelopes the predefined shape utilized in section 3.1 and 3.2, and another one is the same as the binder outline. Both one-layer 45° and two-layer 0°/45° preforming simulation results in (1) Fig. 12 for geometry and (2) Table 1 and Table 2 for size and surface area show that optimal blank geometries that are the same as the ones in Section 3.1 and 3.2, except for small errors caused by mesh difference, can be obtained after one iteration. However, for more complex parts in applications other than automobile industry, design following the flowchart in Fig. 4 is still essential to ensure that the optimal blank geometry is obtained. Fig. 12 Quarter-view of starting blank geometry (green), optimal blank geometries of 45 ° single-layer (red), and 0 ° (yellow) / 45 ° (blue) doublelayer preforming: (a) rectangular starting geometry; and (b) binder outline starting geometry. 
This paper reports a newly developed integrated approach combining FEA modeling method and automatic mesh adjustment to optimize initial blank geometry of the prepregs for preforming processes. The simulation results show that this approach can serve as a reliable foundation for initial blank geometry design even when trial-and-error optimization is applied. Potential benefits of this approach include significant reduction in raw material consumption and trimming cost. Since the design work is performed using FEA models on computers when this approach is applied, raw material consumption from physical trial-and-error preforming tests during the design stage can also be scaled down greatly. In the future, experiments with various process conditions, such as part geometries and stacking configurations, are planned to further validate this integrated approach. More advanced optimization algorithms will also be developed based on this approach to reduce cost of computation and element-byelement mesh adjustment. Moreover, advanced mesh adjustment techniques are to be introduced to solve the jagged edge issue from the root and further lower the trimming requirement.