Structural design of high-performance garments based on geometric optimization

The main goal of performance apparel is to enhance performance through functional design for different sports.

Firstly, enhancing performance means that performance sportswear should help to increase the speed, endurance and agility of people, for example, running or cycling clothing is often designed to be tight-fitting in order to reduce air resistance and improve movement efficiency. Secondly, in terms of reducing discomfort and risk of injury during exercise, the materials used in high-performance sportswear need to have good tear and abrasion resistance, especially in extreme sports or outdoor adventure activities, where tough materials can protect people from friction or accidental tears, avoiding damage to the garment and reducing the risk of abrasions.

The structural design of high-performance clothing must be fully integrated into the principles of ergonomics, in-depth understanding and analysis of the human body’s movement patterns, posture, and physiological needs [34]. Designers need to study the limb activities of personnel in different sports states, accurate cutting and sewing to reduce the sense of constraints, especially in the elbow, knee and other joints, a reasonable structural design can provide personnel with the necessary support, so that they can maintain stability in the rapid change of direction or intensity training.

The adaptability and adjustment function of the garment are also crucial in the design, i.e. The design of high-performance sportswear should take into account the differences in body shape and individual characteristics of the wearer, to ensure that the garment maintains a good fit on people with different body shapes. At the same time, adjustable features (such as zippers, Velcro and elastic structure design) allow personnel to adjust the fit and comfort of the garment at any time during the exercise process, reducing the psychological burden caused by inappropriate clothing, thus enabling personnel to focus more on athletic performance.

The structural design of the garment should be adapted to the material properties to achieve optimal functionality and comfort. Specifically:

Firstly, when using high elasticity materials, the design should take into account their ductility and responsiveness, and choose suitable sewing techniques and seam layouts to reduce friction, so as to ensure that the garment can provide sufficient support and not produce a sense of constriction when the personnel are playing various types of sports. For the special requirements of different sports, designers need to adjust the shape of the structure and the sewing method, so that the garment can give full play to the performance advantages of the material under the premise of ensuring the freedom of movement.

Secondly, the coordination between material selection and garment structure is also reflected in the consideration of environmental adaptability. Different sports scenes and climatic conditions require clothing with corresponding functionality, such as outdoor sports clothing needs to have strong windproof, waterproof and breathable performance, which requires the material not only to be lightweight, but also has a good barrier and thermoregulation. The structural design of the garment needs to be matched with the functional materials, and scientific cutting and layering design should be adopted to ensure that the garment provides protection while maintaining proper ventilation and comfort. For high temperature and humidity environments, ventilation holes, mesh structures, and adjustable openings can improve air circulation and help personnel maintain a proper body temperature during exercise.

The SIMP method is the most widely used topological optimization method. A general structural optimization problem consists of a design variable x=(x1,x2,⋯) , an objective function F(x), and constraints describing the problem, and the goal of the optimization iterations is to find the minimal value of F(x). The design variable in the SIMP method is generally the density of structural cells, with a value of the density of each cell ranging from 0 to 1 [35]. The mathematical relationship between the design variable (cell density) ρ_e and the material properties (cell Young’s modulus) E_e can be expressed as: (1)Ee(ρe)=Emin+ρep(E0−Emin),ρe∈[0,1]

where E_min represents the Young’s modulus of the null material as a very small non-zero value, and exists to prevent singularities in the stiffness matrix of the performance analysis equations. E₀ is the solid material unitary modulus of elasticity for ρ_e = 1, and p is a penalty factor (usually set to p = 3) to accelerate the 0-1 binarization of the design variables.

For the classical minimum flexibility topology optimization problem, the mapping between the objective function and the design variables and constraint functions can be expressed as the following optimization model: (2)Min C(ρ)=UTKU=∑e=1NEe(ρe)ueTkeue Subject to{ KU=F V(ρ)V0≤VF 0≤ρe≤1

Where C is the structural flexibility (objective function), U is the global displacement vector, K is the overall stiffness matrix, N is the number of discrete cells in the design domain, k_e is the stiffness matrix of cell e, u_e is the displacement vector corresponding to cell e, F is the global external force vector, V(ρ) is the volume of the material during the optimization process, V₀ is the volume of the design domain, and VF is the fraction of the volume that is constrained by the designer.

In order to solve the optimization problem in the above equation, structural optimization is solved using gradient-based optimization methods such as optimization criterion (OC) method, moving asymptote method and most rapid descent method. In addition to this, there are also non-gradient methods such as particle swarm algorithm. Thanks to the simplicity and efficiency of the OC method, the SIMP method usually uses the OC method to update the design variables with the help of an illuminating algorithm for its iterative process: (3)ρenew={ max(0,ρe−m)ifρeBeη≤max(0,ρe−m) min(1,ρe+m) ifρeBeη≥min(1,ρe+m), ρeBeηotherwise

where m is the optimization step (m>0) , η is the damping coefficient (usually set to η=12 ), and B_e is calculated by the following equation: (4)Be=−∂C∂ρeλ∂V∂ρe

where the Lagrange multiplier λ is used to keep the volume of the structure in compliance with the constraints, which is generally computed by the dichotomy method.

For the minimum flexibility problem with volume constraints, the gradients are the sensitivities (derivatives) ∂C∂ρe and ∂V∂ρe of the optimization objective C and volume V, respectively, to the optimization design variables, which are calculated by the following equation: (5)∂C∂ρe=−p(ρe)p−1(E0−Emin)ueTkeue (6)∂V∂ρe=Ve

where V_e is the volume of cell e and the volume sensitivity is ∂V∂ρe=1 when all cell sizes in the design domain are 1 × 1 × 1.

The IGA proposed by Hughes, a famous American mechanic, describes the physical field with the help of the most commonly used curved surface expression in CAD and CG systems, i.e., the B-spline/NURBS curved surface, which realizes the consistent model expression in CAD and CAE [36]. Considering the validity of the computation, the NURBS basis functions are generally defined using recursion: the ordered node vector Ξ=[ξ1,ξ2,⋯,ξn+p+1] has a monotonically non-decreasing property, where the series n is the number of NURBS basis functions (i.e., the number of control points), ξ_i is called a node, and p is the order of the NURBS, and the maximum number of repeating nodes of the node vector is p + 1. For a node vector of open form Ξ, B_i,p(ξ) denotes its corresponding ith univariate B spline basis function as shown in equation (7). (7)Bi,0(ξ)={ 1 If ξi≤ξ<ξi+1 0 Otherwise Bi,p(ξ)=ξ−ξiξi+p−uiBi,p−1(ξ)+ξi+p+1−ξξi+p+1−ξi+1Bi+1,p−1(ξ),(p>0)

The above recursive formula is the Cox-deBoor recursive formula, where the convention 0/0 = 0 is that all node vectors used in this paper are in open form, i.e., the first and last two nodes have a p + 1 degree of repetition.

Introducing positive weight values ω_i for each B spline basis function, the univariate NURBS basis function is defined as: (8)Ni,p(ξ)=Bi,p(ξ)ωi∑j=1nBj,p(ξ)ωj

The NURBS basis functions have many good mathematical properties, the more important ones such as (1) nonnegativity, N_{i, r} ≥ 0∀i; (2) local support, Ni,r(ξ)=0∀ξ∉[ξi,ξi+r+1) ; and (3) higher-order differentiability, and the basis function N_{i, r}(ξ) is differentiable of order r − k, where k is the repetition degree of the ith node.

A NURBS curve of order p can be defined as: (9)C(ξ)=∑i=0nNi,p(ξ)ωiPi∑i=0nNi,p(ξ)ωi,a≤u≤b

where P_i is the ind control point, ω_i is the nonnegative weight corresponding to the control point, and N_i,p(ξ) is the NURBS basis function corresponding to the nonperiodic node vector Ξ of the form: (10)Ξ={a,⋯,a︸p+1,ξp+2,⋯,ξn−p−1,b,⋯,b︸p+1}

Further, a NURBS surface is constructed via a tensor product form using bivariate NURBS basis functions. According to tensor product surface theory, a NURBS surface expressed p times in the u direction q times in the v direction is as follows: (11)S(u,v)=∑i=0n∑j=0mNi,p(u)Nj,q(v)ωi,jPi,j∑i=0n∑j=0mNi,p(u)Nj,q(v)ωi,j

where control point P_i,j is tensed into a two-dimensional grid of control points, and N_i,p(u) and N_j,q(v) are basis functions in two directions corresponding to node vectors U and V. Will be bivariate NURBS basis functions: (12)Ri,j(u,v)=Ni,p(u)Nj,q(v)ωi,j∑k=0n∑l=0mNk,p(u)Nl,q(v)ωk,l

Introducing Eq. (10) simplifies the NURBS surface expression to: (13)S(u,v)=∑i=0n∑j=0mRi,j(u,v)Pi,j

The parameter space of this surface S(u, v) is [u1,u2,⋯,un+p+1]×[v1,v2,⋯,vm+q+1] .

Using B-splines/NURBS as shape functions, IGA is realized by the following procedure. In the minimum flexibility TO problem, the cell stiffness matrix k_e is calculated by the following equation: (14)ke = ∫ΩeBTDBdΩn = ∫Ω^eBTDB|J1|dΩ^n& = ∫Ω¯eBTDB|J1|J2|dΩ¯

where Ω_e is the cell e region of the solid domain, B is the strain-displacement matrix, D is the stress-strain matrix, Ω^e is the cell e region of the NURBS parameter space, and Ω¯e is the parent cell e region of the integral domain. Where the Jacobi transformation matrices from the parameter domain to the solid domain and from the Gaussian integral domain to the parameter domain are J₁ and J₂, respectively.When only the 2D stress problem is considered, the B matrices are computed by the following equation: (15)B=[ ∂N1∂x 0 ⋯ ∂Nn∂x 0 0 ∂N1∂y ⋯ 0 ∂Nn∂y ∂N1∂x ∂N1∂y ⋯ ∂Nn∂x ∂Nn∂y] (16)[ ∂Ni∂x ∂Ni∂y]=[ ∂Ni∂u ∂Ni∂v]J1−1

where N_i is the NURBS basis function in the parameter domain and the Jacobi matrix J₁ from the parameter domain to the solid domain is computed by the following equation: (17)J1=[ ∂xi∂u ∂yi∂v ∂xi∂v ∂yi∂v]

In addition, the mapping from integral domain [− 1, 1] to parameter domain [u,ui+1)×(v,vj+1] is shown below: (18){ u=ui+1−ui2(u¯−1)+ui v=vj+1−vj2(v¯−1)+vj

So the corresponding Jacobi matrix J₂ is calculated by the following equation: (19)J2=[ ui+1−ui2 0 0 vj+1−vj2]

The input to this method consists of a template grid of garment structures T , a target garment structure R , and a sparse correspondence C={(t1,r1),…,(tK,rK)} as a constraint, where K is the number of corresponding vertices in the constraint. We first design the template mesh T and integrate it into the algorithmic framework, and the corresponding constraint C can be automatically taken by the algorithm in Chapter 3 or manually specified by the user. The following section describes the core algorithm of optimization and explains the specific implementation details. The algorithm and optimization flow in this chapter are shown in Fig. 1.

Figure 1.

This chapter algorithm and optimization process

The algorithm in this chapter actually uses an optimization method to drive the template mesh to deform into the target garment structure, and does not require the same number of vertices and triangles between the template and the input target garment structure, nor does it require that they have the same topology, which is referred to as a “hard constraint” on the sparse correspondence.

For each triangle, a fourth vertex needs to be added in the space perpendicular to the triangle. Specifically, let v_i and v˜i , i = 1…3, denote the coordinates of the vertices of the triangle before and after the deformation, respectively, then the fourth vertex before the deformation is: (20)v4=v1+(v2−v1)×(v3−v1)|(v2−v1)×(v3−v1)|

The calculation of v˜4 is similar to v₄. The fourth vertex is actually the normal to the face of that triangle. The affine transformation for each triangle vertex can be written as: (21)Tvi+d=v˜i

This can be obtained by substituting each of the four vertices into Eq. (21) and eliminating d: (22)T=V˜V−1

where V=[v2−v1,v3−v1,v4−v1] , V˜=[v˜2−v˜1,v˜3−v˜1,v˜4−v˜1] . Equation (22) is the non-translational part of the affine transformation. Then the smoothing term E_S can be defined as: (23)ES(v1…vn)=∑i=1|T|∑j∈N(i)‖Ti−Tj‖F2

where N(i) denotes the set of triangles adjacent to the ind triangle, ‖⋅‖F denotes the Frobenius norm of the matrix, and v_i is the vertex on the template. This term implies that the deformations of neighboring triangles should be as equal as possible.

The purpose of E_I is to prevent E_S from generating violent jitter by constraining it using the unit matrix I, i.e., when E_I is smallest, all changes approximate the unit matrix. Thus, E_I can be defined as: (24)EI(vi…vn)=∑i=1|T|‖Ti−I‖F2

The last error term, E_V, indicates that each vertex on the template mesh should approximate the nearest valid point on the target garment structure after deformation, similar to measuring the point-by-point error after deformation, which can be defined as: (25)EV(v1…vn,r1…rn)=∑i=1n‖vi−ri‖2

where r_i denotes the nearest valid point on the template grid to the target garment structure and v_i is the template grid vertex. The normal direction is used here to determine the validity of the nearest point. Specifically, a point is determined to be valid if the angle between the normal of a vertex on the template mesh and the normal of the nearest triangle on the target garment structure is less than 90°. The angle may be calculated simply by vector fork product. In the implementation, the search process for the nearest point can be accelerated by KD-Tree.

Finally, Eq. (23), Eq. (24) and Eq. (25) are combined and the deformed template vertices (v˜1,…,v˜n) are computed by minimizing the following optimization problem, viz: (26)minv˜1…,v˜nE(v1…vn,r1…rn)=αES+βEI+γEV s.t.v˜k=rk,k∈1…K

where α, β and γ are the weighting parameters. In this chapter, an iterative strategy is adopted to solve Eq. (26). In each iteration, it is necessary to solve a linear system which can be quickly solved by matrix factorization.

From the technical point of view alone, the methods in this chapter can deform an arbitrary 3D garment structure model into a target garment structure, e.g., deform a male model into a female model. However, in order to facilitate related applications such as garment design, we provide a template grid of male and female garment structures in standard poses beforehand. Meanwhile, to reflect the diversity of available templates, this chapter selects template grids from different datasets. Specifically, the undeformed initial pose from the SCAPE dataset was used as the male template, and a moderate body type was selected from the female clothing structure of the SPRING dataset as the female template. In addition, in order to make the reconstructed 3D garment structure mesh have a relatively regular topology, we first reclassify the model mesh by deleting the vertices and edges of the non-2D manifolds present on the template mesh (which is not the target garment structure) and filling in the holes on the mesh surface. The method reclassifies the triangular mesh into an isotropic mesh surface by computing the orientation field aligned with the surface features. The resulting mesh can have a regular topology.

For sparse correspondences, we extracted 22 corresponding key points from the template garment structure and the target garment structure, respectively: left and right shoulder points, left and right armpit points, two breast tip points, two elbow points, waist points, two knee points, two medial ankle points, one front neck point, one cervical vertebrae point, and two lateral neck points. The extracted keypoints may be intersections of rays with the grid, i.e., the extracted keypoints may not be vertices in the original grid.

The algorithms in this chapter require the input scanned garment structures to be represented as a mesh. For the point cloud of the garment structure obtained by scanning with the depth camera, it needs to be noise-reduced and encapsulated into a mesh first. The designed scanning system acquires the real scanned garment structures, which can automatically complete the basic geometric processing operations such as point cloud noise reduction, alignment, encapsulation, etc., and directly output the 3D garment structure mesh. In addition, as with the template processing, all target garment structures are scaled to the unit sphere, and the real size can be recovered according to the scaling factor after the reconstruction is completed.

In this chapter, the iterative approach of Eq. (26) is also investigated. In the first iteration, by setting γ = 0 to ignore E_M, and setting α = 0.1 and β = 0.01, the vertices on the template mesh that exist within the correspondence are forced to move to the corresponding positions on the target garment structure due to the restriction of the sparse correspondence specified as “hard constraints”, which ensures that there is a correspondence between the key points of the template and the reconstructed garment structure. Other vertices and triangles on the template will be optimized according to E_S and E_I. The vertex positions after the first iteration can be used to heuristically estimate the effective nearest points for subsequent iterations. For subsequent iterations, γ is incrementally added and the effective nearest point is updated after each iteration. In the specific implementation, the value of γ is increased from 1 to 10 through 10 iterations while keeping α and β constant.

The 3D garment structure method based on isogeometric optimization in this section analyzes the parametric scheme of the petal-like tensile expansion structure, which requires only 8 design variables (l1,l2,l3,l4,h4,d1,d2,d3) to describe the geometry as shown in Fig. 2. For simplicity, the width of the connecting strip d₃ is set equal to the petal width d₂ and only one material is considered in the cell element. As a result, the number of design variables used for the petal configuration is reduced to x=[l1,l2,l3,l4,h4,d1,d2] . The tensile-expansive structure RVE for three, four and six petals is shown in Fig. 2. Where (a) is the internal boundary definition; (b) is the external boundary generation; (c) is the parent structure generation; (d) is the insertion of the connecting rod with the transformation of the petal; and (e) is the complete RVE generation.

Figure 2.

Isogeometric parameterization scheme

In order to verify the accuracy and validity of the methodology of this paper, three different petal-like tension expansion structures are tested in this section as shown in Fig. 3.

Figure 3.

Three, four, and six petals of a petal form

Figure 3(a) shows a three-petal structure style with design variables [2.42,0.50,2.50,0.25,10.00,0.30, 1.00], Figure 3(b) a four-petal structure with design variables [1.5,1,1,0.5,10.25,0.5,0.5], and Figure 3(b) a six-petal structure with design variables [1.64,0.85, 1.02,0.61,8.29,0.67,0.98]. The computational time and equivalent mechanical properties results for the isogeometric homogenization calculations and the method of this paper are shown in Table 1.

In the table, the computation time of this paper’s method is much smaller than that of the isogeometric homogenization calculation. In addition, the equivalent Poisson’s ratio and Young’s modulus obtained by this paper’s method are approximately the same as the results of the isogeometric homogenization calculations, with a difference of only 0.1 between the two. Therefore, the efficiency and accuracy of the method in this paper are proved.

In order to find the design limit of the petal-like tension-expansion structure, i.e., the lowest Poisson’s ratio achievable for a given stiffness constraint, the objective function of the shape optimization, i.e., the objective equivalent Poisson’s ratio, can be set to ν = -1. The upper and lower bounds of the design variables in the petal configuration are the same as those of the validation set in the previous section. Two cases are explored in this section: case A for the isogeometric homogenization calculation and case B for the method of this paper.Using these two different methods, the design limit curves for the three- and six-petal tensile expansion structures are shown in Fig. 4. Each data point on the curve in the figure is the lowest Poisson’s ratio for each petal-shaped tension-expansion configuration under stiffness constraints, and the lowest Poisson’s ratio of the design limit curves obtained by this paper’s method is closer to the lowest Poisson’s ratio of the isogeometric homogenization calculation, which proves its accuracy. At lower or higher effective stiffness ranges, the petal-shaped structures lose their tensile expansion properties.

Figure 4.

The design limit curve of this method

As the human skin is always breathing, gas exchange with the outside world, only then, the human body will not feel stuffy and uncomfortable, so the breathability of clothing materials is a key indicator, and breathability is also conducive to the dissipation of body heat. For the breathability of clothing, the thickness and density of the fabric are the main factors affecting its performance. Table 2 displays the results of the garment’s conventional characteristics.

From the data in the table, it can be seen that the breathability of the test suit (0.0644) is much better than the individual materials, so when the subject generates body heat in a cold environment, the heat will be easily transmitted outward, which favors the water portion of the heat to be transmitted outward to keep the body’s body surface dry and to prevent the dissipation of excessive heat. Although the breathability of the test suit is not conducive to the protection of heat, the high resistance to breathability of the outer fabric provides good protection against the continued outward dissipation of heat and creates an effective layer of hot air between the body and the fabric, thus enabling effective warmth and protection from the cold.

The contact angle of the fabric surface determines its hydrophobicity and stain resistance. The higher the contact angle, the stronger the hydrophobicity and dirt resistance, and the water droplets tend to be more spherical. Dirt is a major factor affecting the warmth of fabrics, especially the opening structure of garments such as collar and cuffs, which is the most easily contaminated part of garments due to multiple contaminations such as impurities from the external environment, human secretions and perspiration, and also frequent friction with the skin, so the opening structure is most prone to lose a certain degree of warmth due to dirt. The results of the material moisture characterization tests are shown in Table 3. Where OMMC is the integrated moisture management capacity. As can be seen from the table data, the contact angle of the material is large, which means that the material has obvious hydrophobic, anti-dirt function, especially conducive to preventing the external rain and snow erosion of the clothing - human body’s internal environment, and to maintain a dry environment inside the human body and the clothing, so the traditional clothing structure design of the clothing is suitable for the fabric of the cold-proof clothing, especially suitable for use in the rain and snow environment of the warm clothing.

Secondly, the moisture permeability of the test suit is better than that of the traditional clothing structure, 1782.855>960.767. This is because when the human body carries out outdoor sports in a low-temperature environment, once the human body sweats after the exercise, it is necessary that only the sweat is quickly discharged to keep the human body skin dry to prevent the heat from dissipating with the transmission of humidity moisture well, and ultimately to prevent a rapid drop in the temperature of the skin, to Maintain the thermal comfort of the human body.

Table 4 shows the results of the clothing hydraulic conductivity test. From the table, it can be seen that the moisture permeability time of the material is much larger than that of the test suit, the moisture permeability time of the bottom surface of the material is much larger than that of its surface, and the longest time of the surface moisture permeability is 13.541, which can be inferred that the human body’s sweat can be easily dispersed through the test suit, but the water vapor of the external environment is not easy to enter into the inner environment of the garment through the material. The data in the table also shows that the radius of moisture absorption of the material is smaller than that of the test suit, and the radius of moisture absorption of the surface of the material is much larger than that of its bottom surface; the speed of water transfer of the material is larger than that of the test suit, and the speed of water transfer of the surface of the material is larger than that of its bottom surface. All of these can also show that sweat can be easily dispersed through the test suit, and water vapor from the external environment cannot enter the inside of the garment through the material.

The KES Fabric Style Meter can separately determine 16 physical quantities used to assess hand style. It can provide important information to fabric and garment manufacturers, and provide an effective means of quality control and assurance for the automation of production in the textile and garment industries. Table 5 shows the data results of the KES clothing style evaluation test indicators.

The data in the table shows that the values of bending stiffness and shear hardness of the material are greater than the test fitting, which indicates that the outermost material has good resistance to abrasion, stretching and tearing. A material with good abrasion and tear resistance can meet the needs of outdoor sports where there is a lot of movement and contact with the outside world, such as scratching and grinding. The bending stiffness and shear hardness of the test suit have the smallest values, which fully demonstrates that the test suit has good softness, which meets the needs of the inner layer of clothing close to the wearer’s body.

FIGS. 5 through 8 show the performance of bending, stretching, shearing, and garment characterization results for a conventional garment structural design garment. Figures 9 to 12 show the results of bending, tensile, shear, and test suit characterization for the test suit.

Figure 5.

Tensile ability of materials

Figure 6.

Shear of material

Figure 7.

Bending of material

Figure 8.

Surface characteristics of materials

Figure 9.

Tensile of test loading

Figure 10.

Test load shear

Figure 11.

Surface characteristics of test loading

Figure 12.

Bending of test loading

As can be seen from the figures, the compression linearity values of the material and the test suit are similar, and the compression ratio of the test suit is greater than that of the material, i.e., the fluffy feeling of the garment based on the isogeometric optimization of the garment structural design is better than that of the traditional garment structural design, and the friction between the skin of the wearer and the test suit during the wearer’s activities will not cause discomfort.

From the overall performance point of view, the performance of garments designed by the optimized garment structure design method proposed in this paper is better than that of garments designed by the traditional garment structure design method.