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An optimization procedure for spot-welded structures based on SIMP method
Computational Materials Science xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
An optimization procedure for spot-welded structures based on SIMP method Haiqiang Long a,⇑, Yumei Hu a,b, Xiaoqing Jin a, Huili Yu b, Hao Zhu a a b
State Key Laboratory of Mechanical Transmissions, Chongqing University, Chongqing 400044, China State Key Laboratory of Vehicle NVH and Safety Technology, Chongqing 400039, China
a r t i c l e
i n f o
Article history: Received 9 June 2015 Received in revised form 27 August 2015 Accepted 30 August 2015 Available online xxxx Keywords: Optimization Spot weld SIMP FEA
a b s t r a c t This paper proposes an optimization procedure for spot-welded structures. The welded sheets are connected through continuous solid materials instead of traditional discrete elements, and a topology optimization with Solid Isotropic Microstructure with Penalization (SIMP) method is used to remove the redundant connecting material. The current optimization procedure is able to predict an optimal number and distribution of spot welds for spot-welded structures. Applications to the double-hat welded crossbeam and D pillar joint of a body-in-white (BIW) show that the proposed optimization procedure furnishes good guidelines to redistribute spot welds in the spot-welded structures. The spot welds optimization of BIW is carried out on a finite element model. In order to verify the model, finite element analysis (FEA) and rig test of torsional stiffness are performed on the BIW, where both the FEA and rig test are carried out under the same boundary conditions. A satisfactory agreement between the FEA and experiment demonstrates the capabilities and the effectiveness of the present method. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction Resistance spot welding is a thermos-electric process where overlapping sheets are positioned between water-cooled electrodes for a short period of time. The resistance to the flow of current through the weld pieces at the pressure point creates heat, leading to a localized melting and weld nugget generation. This technology plays an important role in fabricating sheet metal parts in many products such as automobiles, electronics and airplanes. In automotive engineering, for example, thousands of resistance spot welds (RSWs) are used to assemble the body-in-white (BIW) for vehicles [1], and transfer loadings throughout the structures during bending, torsion and crash. Therefore, the quality of spot welds is crucial for the performance of spot-welded structures. Resistance spot welding is a complicated process, and its comprehensive mechanism is highly interdisciplinary, involving the interactions of such phenomena as electrical, thermal, mechanical and metallurgical. Hence, the welding parameters such as welding current, resistance of welding components, pressure, welding time and holding time will all influence the performance of welding joints [2,3]. In the last few decades, extensive efforts have been paid to study the performance of RSWs, such as strength, fatigue
⇑ Corresponding author. Tel.: +86 138 8351 2407.
and crashworthiness of single spot welded samples [4–7]. But for complicated spot-welded structures, e.g., an assembly of BIW, which is welded with thousands of RSWs, the numbers and distributions of spot welds are important factors affecting the static and dynamic behaviors of the BIW. Therefore, identifying the distributions of spot welds and minimizing the spot welds numbers of spot-welded structures are crucial to promoting the quality of spot-welded structures and reducing the cost of the assembly process. A few studies have been conducted to investigate the spot welds numbers and distributions for spot-welded structures. Zhang and Taylor [8] proposed an optimization algorithm for a two-spot welded plate sample, allowing for modifying the stiffness of the structure so as to maximize the fatigue life. Bhatti [9] proposed an adaptive optimization procedure for spot-welded structures, which iteratively added and removed spot welds to determine the optimal distribution as well as the numbers of spot welds. Ertas [10] proposed a methodology to find the optimal locations of spot welds and the optimal overlapping size of the joined plate for maximum fatigue life through a total strain life formulation. In general, the optimization methods above are based on discrete variables. These methods give the optimal spots numbers or distributions for a spot-welded structure by removing or adding spot welds repeatedly. However, those processes usually involve
E-mail address: [email protected] (H. Long). http://dx.doi.org/10.1016/j.commatsci.2015.08.058 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.
Please cite this article in press as: H. Long et al., Comput. Mater. Sci. (2015), http://dx.doi.org/10.1016/j.commatsci.2015.08.058
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H. Long et al. / Computational Materials Science xxx (2015) xxx–xxx
complicated calculations demanding high computational cost. Based on topology optimization, this paper proposes an optimization procedure to obtain the optimal material layout of structural components. Spot welds are represented by continuous connecting material instead of discrete spots, so that the design variables in the optimization become continuum variables. The present algorithm excels in illustrating the distribution layout of the spot welds with reduced computational cost. The paper is organized as follows. The theoretical fundamentals of topology optimization are first reviewed in the next section, where the implementation of the Solid Isotropic Microstructure with Penalization (SIMP), sensitivity analysis and approximation are introduced. Section 3 presents the detailed FEA results and their counterparts obtained from the experiments. Finally, the advantages and implications of the present optimization procedure are discussed and the concluding remarks presented.
Fig. 1. For a unit volume material, the volume of void cell is described with dimensions of a, b, c. In the topology optimization process, the material is dominated with void cells, the material can be removed.
2. Optimization method 2.1. Optimization problem
2.2. Sensitivity analysis
An optimization problem includes three factors, that is, design variable, objective function and constraint equation. Generally, the optimization problem can be described mathematically as:
Gradient based optimization is the one of most efficient approaches for topology optimization. In order to perform the gradient analysis efficiently, the adjoint sensitivity analysis method [15] is recalled. The stiffness, displacement and load applied of a structure can be presented as Eq. (4).
8 > < Minimize : Subject to : > :
f ðYÞ ¼ f ðy1 ; y2 ; . . . ; yN Þ g j ðYÞ g Uj 6 0 yLi
6 yi 6
yUi
j ¼ 1; 2; . . . ; M
ð1Þ
i ¼ 1; 2; . . . ; N
wherein the present study, the design variable Y ¼ ðy1 ; y2 ; . . . ; yn Þ is the volume of the connecting elements, and the displacement corresponding to the loadings is the design response. The displacement is restrained by the upper bound of g U : The objective function f(Y) represents the total volume of connecting material. Derived from Eq. (1), the optimization procedure is seeking the minimum connecting material volume under the responding displacement constraint condition. The design variables in this paper characterize the generalized material distribution allowing topological changes in the design domains. In the literatures, there are a number of topology optimization methods such as homogenization method [11], artificial material method [12,13], and evolutionary structural optimization [14], to name a few. Solid Isotropic Microstructure with Penalization (SIMP) method [12] probably is one of the most widely used artificial material methods. This method is employed in the present study, where the key idea is that the stiffness of the material with penalty is assumed to be:
K p ðqÞ ¼ qp K
ð2Þ
where K P and K represent the penalized and the real stiffness matrix of an element, respectively, q is the artificial density and p the penalization factor. For a unit cube of material containing a given cuboidal void of size (a, b, c), where a, b, and c denote the side lengths of the cube (Fig. 1), the artificial material density is calculated in the non-dimensional sense as:
q¼1abc
ð3Þ
where the artificial material density, q, is a dimensionless value. The continuum hypothesis of the material virtually assumes that there are an infinite number of such voided cells in the material. For every single element, there is nearly no material in the element if the density of material approaches to 0; while if the density of material approaches to 1, it indicates there is no void in the solid element.
½KfUg ¼ fPg
ð4Þ
where K is the stiffness matrix of the structure, and P is the load vector applied. Differentiating the both sides of Eq. (4) with respect to y yields
K
@U @P @K ¼ U @y @y @y
ð5Þ
The constrained g is calculated from the displacements as
g ¼ fQ gT fUg
ð6Þ
The gradient of the constraint is
@g @Q T @fUg ¼ fUg þ Q T @y @y @y
ð7Þ
In the adjoint method of sensitivity analysis, the adjoint displacement vector is designated as E, which satisfies
½KfEg ¼ fQ g
ð8Þ
By substituting Eqs. (5) and (8) into (7), the derivative of the constraints can be represented as:
@g @Q T @fPg @½K ¼ fUg fUg þ fEgT @y @y @y @y
ð9Þ
In Eq. (9), a single forward backward substitution is needed for each retained constraint. The adjoint method of sensitivity analysis can be used hereby to reduce the computational costs. 2.3. Approximation formulation The general method of solution to the optimization problem as shown in Eq. (1) is through the approximation approach. In this approach, the optimization problem is tackled by solving a series of explicit approximate problems. The overall efficiency of this approach is determined by the accuracy of the approximation. Three typical approximation formulations used in structural optimization are:
Please cite this article in press as: H. Long et al., Comput. Mater. Sci. (2015), http://dx.doi.org/10.1016/j.commatsci.2015.08.058
H. Long et al. / Computational Materials Science xxx (2015) xxx–xxx
g~j ðyÞ ¼ g j0 þ
N X @g j i¼1
g~j ðyÞ ¼ g j0
@yi
N X @g j i¼1
@yi
ðyi yi0 Þ y2i0
1 1 yi yi0
ð10Þ
8 N X > @g j > > > cji ðyi yi0 Þ g~j ðyÞ ¼ g j0 þ > > @yi > i¼1 > > 8 < @g j > > > < 1; if @y P 0; > > i > with c ¼ > > ji > > @g j > yi0 > > > > <0 : ; if : yi @yi
ð11Þ
domain is analyzed, which treats the optimization as a sizing problem with the density as the variable. Step 3. With the guidance of the optimization output, a modified structure is modeled and a verification analysis is performed. Based on the verification result, the optimization is repeated till a reasonable result is achieved. In what follows, we present the detailed results of the finite element analysis and the related experimental study.
ð12Þ
It is noted in the literatures of structural optimization, that the above Eqs. (10)–(12), bear the names of linear approximation, reciprocal approximation [16], and convex approximation [17], respectively. In practice, most modern commercial software can pickup the best suited approximation formulations automatically. 3. Numerical and experimental studies In this work, the optimization is performed by commercial software Altair OptiStruct. The overall working scheme is shown in the flowchart (Fig. 2), which contains the following main steps: Step 1. A preprocessor is performed to obtain the finite element model. Instead of traditional discrete spot welds, continuum connecting material is used to connect the welded sheets. A finite element analysis is carried out on the base design and the displacement corresponding to the loads is used to define the optimization objective response. Accordingly, the design variables and optimization domains are defined for optimization. Step 2: Optimization is performed on the structure via OptiStruct. The effective material properties are carried out by the SIMP method. Then, optimal layout of material in design
Fig. 2. Flowchart of the optimization process.
3
3.1. Spot welds optimization for simple structure First, a simple structure of double-hat welded crossbeam is exemplified here. The length of the crossbeam is 1000 mm, with the section dimensions shown in Fig. 3. The Young’s modulus of the material is 200 GPa, Poisson’s ratio 0.3, and the density 7850 kg/m3. The double-hat crossbeam is modeled by both 40 ACM spots and continuum solid connecting welded structure, as shown in Fig. 4a and b, respectively. The local magnified domain (Fig. 4a) describes the model of ACM spot welds, and the other enlarged detail view (Fig. 4b) shows that the shell elements of crossbeam are connected through coincident nodes at the vertices of the solid elements. The continuum connecting material situates in the design domain and redundant material will be removed in the optimization process. Three loading cases (Table 1) are examined, and the corresponding boundary conditions are simplified as shown schematically in Fig. 5. For the first two loading cases, a full constraint is enforced at one end, and a force of 1500 N is applied at the other end of the crossbeam (i.e., nodal numbered 5950). The force is applied in the x (case 1) and y direction (case 2), respectively. In case 3, the welded crossbeam is constrained at both ends, and a force of 1500 N is applied at the middle of the beam in the x direction at node 5951. A tentative analysis is performed to the crossbeam welded with 40 ACM spot welds under the same boundary conditions, in order to obtain the displacements of the critical nodes, which is used as the constraint objectives corresponding to cases in the optimization process. The optimization is carried out under the same boundary conditions for the primitive design and the result is shown in Fig. 6a. The red elements are retained while the other gray elements can be removed. Consequently, after removing the redundant material, Fig. 6b shows the discrete layout of spot welds with only 4% connecting material retained. Following the guidance of the topology optimization, some spot welds may need to be added to reduce the distance between spot welds. The present optimization procedure demonstrates that the double-hat crossbeam can be welded with only 20 spot welds and the maximum corresponding displacement increases only 2.1%, compared to the one welded with 40 ACM spot welds. The optimization reduces the spot welds significantly while the increase of displacement is acceptable. The comparisons of the displacement corresponding
Fig. 3. Dimensions of the hat-shaped crossbeam section (mm).
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Fig. 4. The FE models of double-hat welded crossbeam, a shows the structure modeled with 40 ACM spots and b shows that the upper and lower hat beams are connected continuum solid material which represents the spot welds.
to each loading case are shown in Table 1, where the primitive design stands for the model welded with 40 ACM spot welds, the modified design is the structure welded with continuum solid material, and the optimized design is the one welded with 20 spot welds after topology optimization, respectively. 3.2. Application of spot optimization for BIW Resistance spot welding is widely used in automotive industry, for example, a BIW is assembled by 3000–5000 RSWs or even more. The distribution of spot welds is a key factor that influences not only the fundamental performance of BIW (e.g., torsional and bending stiffness), but also the assembly cost of vehicles. In order to examine the applicability of the proposed optimization procedure for determining the number and distribution layout of spot weld in a BIW, model validation followed by topology optimization based on the torsional stiffness are presented next. 3.2.1. Model validation The torsional stiffness analysis and spot welds optimization are carried out via a finite element model of BIW. Initially, the BIW is assembled with over 5500 spot welds by geometrical designing and modeled via ACM spot welds. In the present work, torsional stiffness of the BIW is studied by the finite element analysis and then compared with the rig test under the same boundary conditions. The FE model of BIW is shown in Fig. 7, where a displacement equation of the left and right front shock tower centers numbered 5010 and 5020 was defined. The equation gives a displacement constraint of equal in size but opposite in direction between node 5010 and 5020 in the z direction. The distance between nodes 5010 and 5020 is 1097 mm, so that a force of
1823 N applied to node 5010 will generate the torsion of 2000 Nm to the BIW. The constraints are applied to the rear spring bumpers at the nodes numbered 5030 and 5040 in the left and right, respectively (cf. Fig. 7). Fig. 8 depicts the 24 measured nodes which are numbered from 1001 to 1012 in the left side and 2001 to 2012 in the right side, respectively. The torsional test of the BIW was performed experimentally as shown in Fig. 9. The left and right rear spring bumpers are welded on the test rig. To support the car, the front shock towers are mounted to a stiff crossbeam, which can rotate around the middle supporting point freely. The constraint points and a simplified force are shown in Fig. 9. The displacements are measured by displacement sensors located at the measuring points. A loading and unloading preprocess is performed to reset the measuring system before the rig test is performed. The rig tests are carried out three times under the same conditions to determine the average displacements and error bands at the points measured. With the torsional moment of 2000 Nm, the averaged displacements and error bands of the left measured points are shown in Fig. 10. Compared with the average value, the displacements of FEA exhibit good agreement with those of the rig test. Also, the displacements at left and right measuring points show a symmetric distribution. The maximum relative deviation between the FEA and test results is less than 10%, therefore, the comparisons validate the FE model for BIW optimization. 3.2.2. Spot optimization of BIW In a BIW structure, the stiffness of joints at pillars is a vital factor which significantly affects the torsional stiffness of the BIW. In order to simplify the preprocessor and reduce the computation cost, spot welds located at the critical locations, e.g., the upper and lower D pillar joints (Fig. 11) are studied in the work. To build the optimal model, firstly, the ACM spot welds which located at the upper and lower D pillar are removed and a refined mesh with averaged size of 3 mm is performed on the connecting sheets at D pillar. It should be noted that the upper and lower connecting sheets are modeled with the same number of elements and the elements should be aligned one-to-one as far as possible (see locations A and B in Fig. 11). Then the continuum solid connecting elements are built, and the solid connecting elements coincided with the shell elements at the vertex nodes. In the topology optimization process, the volume of solid connecting elements is the objective. The displacements of the measured nodes of base design are defined as the optimization constraint and the optimization based on torsional analysis is carried out under the identical boundary conditions of base design.
Table 1 The loading cases and corresponding displacements (applied load = 1500 N). Case #
Measured nodal number
Displacement (mm)
Primitive design
#1 #2 #3
5950 5950 5951
9.13 8.11 0.46
Modified design
#1 #2 #3
5950 5950 5951
9.03 7.41 0.42
Optimized design
#1 #2 #3
5950 5950 5951
9.25 8.26 0.47
Fig. 5. A simplified model defines boundary conditions to the crossbeam in the cases study. In cases 1 and 2, the force is applied to the node numbered 5950 and in case 3, the force is applied to the node numbered 5951. The measured node is the node which the force applied to.
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Fig. 6. The result of topology optimization for a double-hat welded beam. The material density of the red connecting elements approaches to 1, and the density of other gray elements are approximate to 0. The gray elements can be removed and the layout of the removed connecting material distributes discretely as shown in b.
5
Fig. 8. FE model of BIW, the measured nodes are shown here and the 24 measured nodes numbered from 1001 to 1012 in the left side and 2001 to 2012 in the right side respectively.
Fig. 9. Torsional stiffness test of BIW, in the rig test, the measured points are located at the same location as FE analysis.
Fig. 7. The FE model of BIW, a force of 1823 N is applied to the front left shock tower center, the node numbered 5010. Constraints are applied the rear bumper center, the left and right node numbered 5030 and 5040 respectively.
Over 60 iterations are performed in the topology optimization process to achieve a numerical convergence. The result gives the numbers and distribution layouts of the spot welds. In contrast, the original continuum solid and the optimized spot welds are shown in Fig. 12. The red discrete elements are the retained material recommended by the topology optimization. With the guideline, the torsional stiffness of BIW increased by 0.35% through spots redistribution at the upper and lower D pillars regions. Also, it has been demonstrated in another similar study that with the topology optimization, 6 spot welds that is 18% of spot welds at the D pillar regions could be removed, while the torsional stiffness of the BIW virtually keeps constant.
Fig. 10. The averaged displacement and error band of test result of the left measured points. And the comparative result of FEA and rig test.
4. Results and discussions This paper proposes an optimization procedure for spot-welded structures, which applies the SIMP method to perform a topology optimization and find out the optimal distributions and numbers of spot welds in spot-welded structures. In the example of the double-hat welded crossbeam, the welded sheets are connected with continuum material instead of discrete spot welds. Topology optimization is performed on the welded structure, where the design variable is the connecting material. The optimization result
shows that there are over 96% redundant connecting material can be removed. With the guidance of the topology optimization, the double-hat crossbeam can be welded with only 20 spot welds and the maximum displacement corresponding to loading increases by only 2.1% compared to that welded with 40 ACM spot welds. The optimization procedure not only reduces 50% spot welds of the double-hat welded crossbeam, but also relocates spot welds. Compared to the significant reduction of spot welds, the increase of displacement is acceptable.
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the example of torsional stiffness optimization of BIW in this paper. The optimization process is divided into two stages. At the first stage, a general analysis is performed to find out the critical regions that affect the performance significantly. After that, a local refined model with solid connecting elements between welding sheets is built and the solid connecting elements are defined as the topological optimization domain to give a local topological optimization for the structure. The other method is to apply this optimization method in conceptual design phrase of the BIW. It is feasible to use continuum solid material to represent the spot welds in most conceptual CAE models [18], which can be simplified by beams and shell elements. 5. Conclusions
Fig. 11. FE model of solid connection the welded sheets at the upper and lower D pillar of BIW.
This paper proposes a spot welds optimization procedure, where the welded sheets are connected by continuum solid elements instead of discrete elements. The process turns the discrete optimization problem into a continuous one. The connecting material is defined as design variable and topological optimization with the SIMP method is applied to remove the redundant material to obtain the minimum material volume which generates a layout for spot welds. The applications of spot welds optimization for a double-hat welded crossbeam and D pillar joints of a BIW show that the proposed method furnishes a good guideline for distributing the spot welds in spot-welded structures. The proposed spot welds optimization procedure could reduce the spot welds number effectively without noticeably sacrificing the performance of the spot-welded structure. The present study provides a feasible solution to spot-welded structures in promoting the welding quality and reducing cost. Acknowledgements
Fig. 12. The spot welds distributions are recommended by topology optimization result. The whole elements are modeled by continuum solid connecting elements to represent the original spot welds. With the topology optimization, the discrete red elements are recommended as the optimal spot welds for the structure, and the gray elements are redundant materials. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
An optimization is performed to a BIW, and a comparative study of torsional stiffness is conducted to verify the FE model of BIW. The analysis and rig test result are carried out under the same boundary conditions. The present study shows that the maximum error between the averaged test result and FEA is less than 10% in displacement at measuring points. The application to the BIW shows that the optimization method is feasible for complicated welded structures, although it is somewhat nontrivial to build a comprehensive model, such as a BIW, which is assembled with thousands of spots and welded sheets of complicated geometric shape. The present problem is how to efficiently apply the optimization method for the welded structure with a large number of elements or complex geometric structures. There are two potential methods to solve the problem. One of the methods is shown in
This work is supported by the National Natural Science Foundation of China (Grant No. 51475057), Fundamental Research Funds for the Central Universities (No. CDJZR14285501). X.J. would also like to acknowledge the supports from Chongqing City Science and Technology Program (No. cstc2013jcyjA0975). References [1] S.W. Chae, K.Y. Kwon, T.S. Lee, Finite Elem. Anal. Des. 38 (3) (2002) 227–294. [2] B. Bouyousfi, T. Sahraoui, S. Guessasma, K.T. Chaouch, Mater. Des. 28 (2) (2007) 414–419. [3] M. Pouranvari, H.R. Asgari, S.M. Mosavizadch, P.H. Marashi, M. Goodarzi, Sci. Technol. Weld. Joining 12 (3) (2007) 217–225. [4] A. Al-Samhan, S.M.H. Darwish, Int. J. Adhes. Adhes. 23 (1) (2003) 23–28. [5] S. Mahadevan, K. Ni, Reliab. Eng. Syst. Saf. 81 (1) (2003) 9–21. [6] S. Aslanlar, A. Ogur, U. Ozsarac, E. Ilhan, Mater. Des. 29 (7) (2008) 1427–1431. [7] P. Marashi, M. Pouranvari, S. Amirabdollahian, A. Abedi, M. Goodarzi, Mater. Sci. Eng.: A 480 (1) (2008) 175–180. [8] Y. Zhang, D. Taylor, Finite Elem. Anal. Des. 37 (12) (2001) 1113–1122. [9] Q.I. Bhatti, M. Ouisse, S. Cogan, Comput. Struct. 89 (17) (2011) 1697–1711. [10] A.H. Ertas, F.O. Sonmez, Finite Elem. Anal. Des. 47 (4) (2011) 413–423. [11] M.P. Bendsøe, N. Kikuchi, Comput. Meth. Appl. Mech. Eng. 71 (2) (1988) 197–224. [12] M. Bendsøe, Struct. Optim. 1 (4) (1989) 193–202. [13] M.R. Zhou, G.I.N. Rozvany, Comput. Meth. Appl. Mech. Eng. 89 (1) (1991) 309–336. [14] Y.M. Xie, G.P. Steven, Comput. Struct. 49 (5) (1993) 885–896. [15] R.T. Haftka, H.M. Adelman, Struct. Optim. 1 (3) (1989) 137–151. [16] R.T. Haftka, J.H. Starnes, AIAA J. 14 (6) (1976) 718–724. [17] L.A. Schmit, B. Farshi, AIAA J. 12 (5) (1974) 692–699. [18] M.A. Fard, T. Ishihara, H. Inooka, J. Biomech. Eng. 125 (4) (2003) 533–539.
Please cite this article in press as: H. Long et al., Comput. Mater. Sci. (2015), http://dx.doi.org/10.1016/j.commatsci.2015.08.058
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
An optimization procedure for spot-welded structures based on SIMP method Haiqiang Long a,⇑, Yumei Hu a,b, Xiaoqing Jin a, Huili Yu b, Hao Zhu a a b
State Key Laboratory of Mechanical Transmissions, Chongqing University, Chongqing 400044, China State Key Laboratory of Vehicle NVH and Safety Technology, Chongqing 400039, China
a r t i c l e
i n f o
Article history: Received 9 June 2015 Received in revised form 27 August 2015 Accepted 30 August 2015 Available online xxxx Keywords: Optimization Spot weld SIMP FEA
a b s t r a c t This paper proposes an optimization procedure for spot-welded structures. The welded sheets are connected through continuous solid materials instead of traditional discrete elements, and a topology optimization with Solid Isotropic Microstructure with Penalization (SIMP) method is used to remove the redundant connecting material. The current optimization procedure is able to predict an optimal number and distribution of spot welds for spot-welded structures. Applications to the double-hat welded crossbeam and D pillar joint of a body-in-white (BIW) show that the proposed optimization procedure furnishes good guidelines to redistribute spot welds in the spot-welded structures. The spot welds optimization of BIW is carried out on a finite element model. In order to verify the model, finite element analysis (FEA) and rig test of torsional stiffness are performed on the BIW, where both the FEA and rig test are carried out under the same boundary conditions. A satisfactory agreement between the FEA and experiment demonstrates the capabilities and the effectiveness of the present method. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction Resistance spot welding is a thermos-electric process where overlapping sheets are positioned between water-cooled electrodes for a short period of time. The resistance to the flow of current through the weld pieces at the pressure point creates heat, leading to a localized melting and weld nugget generation. This technology plays an important role in fabricating sheet metal parts in many products such as automobiles, electronics and airplanes. In automotive engineering, for example, thousands of resistance spot welds (RSWs) are used to assemble the body-in-white (BIW) for vehicles [1], and transfer loadings throughout the structures during bending, torsion and crash. Therefore, the quality of spot welds is crucial for the performance of spot-welded structures. Resistance spot welding is a complicated process, and its comprehensive mechanism is highly interdisciplinary, involving the interactions of such phenomena as electrical, thermal, mechanical and metallurgical. Hence, the welding parameters such as welding current, resistance of welding components, pressure, welding time and holding time will all influence the performance of welding joints [2,3]. In the last few decades, extensive efforts have been paid to study the performance of RSWs, such as strength, fatigue
⇑ Corresponding author. Tel.: +86 138 8351 2407.
and crashworthiness of single spot welded samples [4–7]. But for complicated spot-welded structures, e.g., an assembly of BIW, which is welded with thousands of RSWs, the numbers and distributions of spot welds are important factors affecting the static and dynamic behaviors of the BIW. Therefore, identifying the distributions of spot welds and minimizing the spot welds numbers of spot-welded structures are crucial to promoting the quality of spot-welded structures and reducing the cost of the assembly process. A few studies have been conducted to investigate the spot welds numbers and distributions for spot-welded structures. Zhang and Taylor [8] proposed an optimization algorithm for a two-spot welded plate sample, allowing for modifying the stiffness of the structure so as to maximize the fatigue life. Bhatti [9] proposed an adaptive optimization procedure for spot-welded structures, which iteratively added and removed spot welds to determine the optimal distribution as well as the numbers of spot welds. Ertas [10] proposed a methodology to find the optimal locations of spot welds and the optimal overlapping size of the joined plate for maximum fatigue life through a total strain life formulation. In general, the optimization methods above are based on discrete variables. These methods give the optimal spots numbers or distributions for a spot-welded structure by removing or adding spot welds repeatedly. However, those processes usually involve
E-mail address: [email protected] (H. Long). http://dx.doi.org/10.1016/j.commatsci.2015.08.058 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.
Please cite this article in press as: H. Long et al., Comput. Mater. Sci. (2015), http://dx.doi.org/10.1016/j.commatsci.2015.08.058
2
H. Long et al. / Computational Materials Science xxx (2015) xxx–xxx
complicated calculations demanding high computational cost. Based on topology optimization, this paper proposes an optimization procedure to obtain the optimal material layout of structural components. Spot welds are represented by continuous connecting material instead of discrete spots, so that the design variables in the optimization become continuum variables. The present algorithm excels in illustrating the distribution layout of the spot welds with reduced computational cost. The paper is organized as follows. The theoretical fundamentals of topology optimization are first reviewed in the next section, where the implementation of the Solid Isotropic Microstructure with Penalization (SIMP), sensitivity analysis and approximation are introduced. Section 3 presents the detailed FEA results and their counterparts obtained from the experiments. Finally, the advantages and implications of the present optimization procedure are discussed and the concluding remarks presented.
Fig. 1. For a unit volume material, the volume of void cell is described with dimensions of a, b, c. In the topology optimization process, the material is dominated with void cells, the material can be removed.
2. Optimization method 2.1. Optimization problem
2.2. Sensitivity analysis
An optimization problem includes three factors, that is, design variable, objective function and constraint equation. Generally, the optimization problem can be described mathematically as:
Gradient based optimization is the one of most efficient approaches for topology optimization. In order to perform the gradient analysis efficiently, the adjoint sensitivity analysis method [15] is recalled. The stiffness, displacement and load applied of a structure can be presented as Eq. (4).
8 > < Minimize : Subject to : > :
f ðYÞ ¼ f ðy1 ; y2 ; . . . ; yN Þ g j ðYÞ g Uj 6 0 yLi
6 yi 6
yUi
j ¼ 1; 2; . . . ; M
ð1Þ
i ¼ 1; 2; . . . ; N
wherein the present study, the design variable Y ¼ ðy1 ; y2 ; . . . ; yn Þ is the volume of the connecting elements, and the displacement corresponding to the loadings is the design response. The displacement is restrained by the upper bound of g U : The objective function f(Y) represents the total volume of connecting material. Derived from Eq. (1), the optimization procedure is seeking the minimum connecting material volume under the responding displacement constraint condition. The design variables in this paper characterize the generalized material distribution allowing topological changes in the design domains. In the literatures, there are a number of topology optimization methods such as homogenization method [11], artificial material method [12,13], and evolutionary structural optimization [14], to name a few. Solid Isotropic Microstructure with Penalization (SIMP) method [12] probably is one of the most widely used artificial material methods. This method is employed in the present study, where the key idea is that the stiffness of the material with penalty is assumed to be:
K p ðqÞ ¼ qp K
ð2Þ
where K P and K represent the penalized and the real stiffness matrix of an element, respectively, q is the artificial density and p the penalization factor. For a unit cube of material containing a given cuboidal void of size (a, b, c), where a, b, and c denote the side lengths of the cube (Fig. 1), the artificial material density is calculated in the non-dimensional sense as:
q¼1abc
ð3Þ
where the artificial material density, q, is a dimensionless value. The continuum hypothesis of the material virtually assumes that there are an infinite number of such voided cells in the material. For every single element, there is nearly no material in the element if the density of material approaches to 0; while if the density of material approaches to 1, it indicates there is no void in the solid element.
½KfUg ¼ fPg
ð4Þ
where K is the stiffness matrix of the structure, and P is the load vector applied. Differentiating the both sides of Eq. (4) with respect to y yields
K
@U @P @K ¼ U @y @y @y
ð5Þ
The constrained g is calculated from the displacements as
g ¼ fQ gT fUg
ð6Þ
The gradient of the constraint is
@g @Q T @fUg ¼ fUg þ Q T @y @y @y
ð7Þ
In the adjoint method of sensitivity analysis, the adjoint displacement vector is designated as E, which satisfies
½KfEg ¼ fQ g
ð8Þ
By substituting Eqs. (5) and (8) into (7), the derivative of the constraints can be represented as:
@g @Q T @fPg @½K ¼ fUg fUg þ fEgT @y @y @y @y
ð9Þ
In Eq. (9), a single forward backward substitution is needed for each retained constraint. The adjoint method of sensitivity analysis can be used hereby to reduce the computational costs. 2.3. Approximation formulation The general method of solution to the optimization problem as shown in Eq. (1) is through the approximation approach. In this approach, the optimization problem is tackled by solving a series of explicit approximate problems. The overall efficiency of this approach is determined by the accuracy of the approximation. Three typical approximation formulations used in structural optimization are:
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H. Long et al. / Computational Materials Science xxx (2015) xxx–xxx
g~j ðyÞ ¼ g j0 þ
N X @g j i¼1
g~j ðyÞ ¼ g j0
@yi
N X @g j i¼1
@yi
ðyi yi0 Þ y2i0
1 1 yi yi0
ð10Þ
8 N X > @g j > > > cji ðyi yi0 Þ g~j ðyÞ ¼ g j0 þ > > @yi > i¼1 > > 8 < @g j > > > < 1; if @y P 0; > > i > with c ¼ > > ji > > @g j > yi0 > > > > <0 : ; if : yi @yi
ð11Þ
domain is analyzed, which treats the optimization as a sizing problem with the density as the variable. Step 3. With the guidance of the optimization output, a modified structure is modeled and a verification analysis is performed. Based on the verification result, the optimization is repeated till a reasonable result is achieved. In what follows, we present the detailed results of the finite element analysis and the related experimental study.
ð12Þ
It is noted in the literatures of structural optimization, that the above Eqs. (10)–(12), bear the names of linear approximation, reciprocal approximation [16], and convex approximation [17], respectively. In practice, most modern commercial software can pickup the best suited approximation formulations automatically. 3. Numerical and experimental studies In this work, the optimization is performed by commercial software Altair OptiStruct. The overall working scheme is shown in the flowchart (Fig. 2), which contains the following main steps: Step 1. A preprocessor is performed to obtain the finite element model. Instead of traditional discrete spot welds, continuum connecting material is used to connect the welded sheets. A finite element analysis is carried out on the base design and the displacement corresponding to the loads is used to define the optimization objective response. Accordingly, the design variables and optimization domains are defined for optimization. Step 2: Optimization is performed on the structure via OptiStruct. The effective material properties are carried out by the SIMP method. Then, optimal layout of material in design
Fig. 2. Flowchart of the optimization process.
3
3.1. Spot welds optimization for simple structure First, a simple structure of double-hat welded crossbeam is exemplified here. The length of the crossbeam is 1000 mm, with the section dimensions shown in Fig. 3. The Young’s modulus of the material is 200 GPa, Poisson’s ratio 0.3, and the density 7850 kg/m3. The double-hat crossbeam is modeled by both 40 ACM spots and continuum solid connecting welded structure, as shown in Fig. 4a and b, respectively. The local magnified domain (Fig. 4a) describes the model of ACM spot welds, and the other enlarged detail view (Fig. 4b) shows that the shell elements of crossbeam are connected through coincident nodes at the vertices of the solid elements. The continuum connecting material situates in the design domain and redundant material will be removed in the optimization process. Three loading cases (Table 1) are examined, and the corresponding boundary conditions are simplified as shown schematically in Fig. 5. For the first two loading cases, a full constraint is enforced at one end, and a force of 1500 N is applied at the other end of the crossbeam (i.e., nodal numbered 5950). The force is applied in the x (case 1) and y direction (case 2), respectively. In case 3, the welded crossbeam is constrained at both ends, and a force of 1500 N is applied at the middle of the beam in the x direction at node 5951. A tentative analysis is performed to the crossbeam welded with 40 ACM spot welds under the same boundary conditions, in order to obtain the displacements of the critical nodes, which is used as the constraint objectives corresponding to cases in the optimization process. The optimization is carried out under the same boundary conditions for the primitive design and the result is shown in Fig. 6a. The red elements are retained while the other gray elements can be removed. Consequently, after removing the redundant material, Fig. 6b shows the discrete layout of spot welds with only 4% connecting material retained. Following the guidance of the topology optimization, some spot welds may need to be added to reduce the distance between spot welds. The present optimization procedure demonstrates that the double-hat crossbeam can be welded with only 20 spot welds and the maximum corresponding displacement increases only 2.1%, compared to the one welded with 40 ACM spot welds. The optimization reduces the spot welds significantly while the increase of displacement is acceptable. The comparisons of the displacement corresponding
Fig. 3. Dimensions of the hat-shaped crossbeam section (mm).
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H. Long et al. / Computational Materials Science xxx (2015) xxx–xxx
Fig. 4. The FE models of double-hat welded crossbeam, a shows the structure modeled with 40 ACM spots and b shows that the upper and lower hat beams are connected continuum solid material which represents the spot welds.
to each loading case are shown in Table 1, where the primitive design stands for the model welded with 40 ACM spot welds, the modified design is the structure welded with continuum solid material, and the optimized design is the one welded with 20 spot welds after topology optimization, respectively. 3.2. Application of spot optimization for BIW Resistance spot welding is widely used in automotive industry, for example, a BIW is assembled by 3000–5000 RSWs or even more. The distribution of spot welds is a key factor that influences not only the fundamental performance of BIW (e.g., torsional and bending stiffness), but also the assembly cost of vehicles. In order to examine the applicability of the proposed optimization procedure for determining the number and distribution layout of spot weld in a BIW, model validation followed by topology optimization based on the torsional stiffness are presented next. 3.2.1. Model validation The torsional stiffness analysis and spot welds optimization are carried out via a finite element model of BIW. Initially, the BIW is assembled with over 5500 spot welds by geometrical designing and modeled via ACM spot welds. In the present work, torsional stiffness of the BIW is studied by the finite element analysis and then compared with the rig test under the same boundary conditions. The FE model of BIW is shown in Fig. 7, where a displacement equation of the left and right front shock tower centers numbered 5010 and 5020 was defined. The equation gives a displacement constraint of equal in size but opposite in direction between node 5010 and 5020 in the z direction. The distance between nodes 5010 and 5020 is 1097 mm, so that a force of
1823 N applied to node 5010 will generate the torsion of 2000 Nm to the BIW. The constraints are applied to the rear spring bumpers at the nodes numbered 5030 and 5040 in the left and right, respectively (cf. Fig. 7). Fig. 8 depicts the 24 measured nodes which are numbered from 1001 to 1012 in the left side and 2001 to 2012 in the right side, respectively. The torsional test of the BIW was performed experimentally as shown in Fig. 9. The left and right rear spring bumpers are welded on the test rig. To support the car, the front shock towers are mounted to a stiff crossbeam, which can rotate around the middle supporting point freely. The constraint points and a simplified force are shown in Fig. 9. The displacements are measured by displacement sensors located at the measuring points. A loading and unloading preprocess is performed to reset the measuring system before the rig test is performed. The rig tests are carried out three times under the same conditions to determine the average displacements and error bands at the points measured. With the torsional moment of 2000 Nm, the averaged displacements and error bands of the left measured points are shown in Fig. 10. Compared with the average value, the displacements of FEA exhibit good agreement with those of the rig test. Also, the displacements at left and right measuring points show a symmetric distribution. The maximum relative deviation between the FEA and test results is less than 10%, therefore, the comparisons validate the FE model for BIW optimization. 3.2.2. Spot optimization of BIW In a BIW structure, the stiffness of joints at pillars is a vital factor which significantly affects the torsional stiffness of the BIW. In order to simplify the preprocessor and reduce the computation cost, spot welds located at the critical locations, e.g., the upper and lower D pillar joints (Fig. 11) are studied in the work. To build the optimal model, firstly, the ACM spot welds which located at the upper and lower D pillar are removed and a refined mesh with averaged size of 3 mm is performed on the connecting sheets at D pillar. It should be noted that the upper and lower connecting sheets are modeled with the same number of elements and the elements should be aligned one-to-one as far as possible (see locations A and B in Fig. 11). Then the continuum solid connecting elements are built, and the solid connecting elements coincided with the shell elements at the vertex nodes. In the topology optimization process, the volume of solid connecting elements is the objective. The displacements of the measured nodes of base design are defined as the optimization constraint and the optimization based on torsional analysis is carried out under the identical boundary conditions of base design.
Table 1 The loading cases and corresponding displacements (applied load = 1500 N). Case #
Measured nodal number
Displacement (mm)
Primitive design
#1 #2 #3
5950 5950 5951
9.13 8.11 0.46
Modified design
#1 #2 #3
5950 5950 5951
9.03 7.41 0.42
Optimized design
#1 #2 #3
5950 5950 5951
9.25 8.26 0.47
Fig. 5. A simplified model defines boundary conditions to the crossbeam in the cases study. In cases 1 and 2, the force is applied to the node numbered 5950 and in case 3, the force is applied to the node numbered 5951. The measured node is the node which the force applied to.
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H. Long et al. / Computational Materials Science xxx (2015) xxx–xxx
Fig. 6. The result of topology optimization for a double-hat welded beam. The material density of the red connecting elements approaches to 1, and the density of other gray elements are approximate to 0. The gray elements can be removed and the layout of the removed connecting material distributes discretely as shown in b.
5
Fig. 8. FE model of BIW, the measured nodes are shown here and the 24 measured nodes numbered from 1001 to 1012 in the left side and 2001 to 2012 in the right side respectively.
Fig. 9. Torsional stiffness test of BIW, in the rig test, the measured points are located at the same location as FE analysis.
Fig. 7. The FE model of BIW, a force of 1823 N is applied to the front left shock tower center, the node numbered 5010. Constraints are applied the rear bumper center, the left and right node numbered 5030 and 5040 respectively.
Over 60 iterations are performed in the topology optimization process to achieve a numerical convergence. The result gives the numbers and distribution layouts of the spot welds. In contrast, the original continuum solid and the optimized spot welds are shown in Fig. 12. The red discrete elements are the retained material recommended by the topology optimization. With the guideline, the torsional stiffness of BIW increased by 0.35% through spots redistribution at the upper and lower D pillars regions. Also, it has been demonstrated in another similar study that with the topology optimization, 6 spot welds that is 18% of spot welds at the D pillar regions could be removed, while the torsional stiffness of the BIW virtually keeps constant.
Fig. 10. The averaged displacement and error band of test result of the left measured points. And the comparative result of FEA and rig test.
4. Results and discussions This paper proposes an optimization procedure for spot-welded structures, which applies the SIMP method to perform a topology optimization and find out the optimal distributions and numbers of spot welds in spot-welded structures. In the example of the double-hat welded crossbeam, the welded sheets are connected with continuum material instead of discrete spot welds. Topology optimization is performed on the welded structure, where the design variable is the connecting material. The optimization result
shows that there are over 96% redundant connecting material can be removed. With the guidance of the topology optimization, the double-hat crossbeam can be welded with only 20 spot welds and the maximum displacement corresponding to loading increases by only 2.1% compared to that welded with 40 ACM spot welds. The optimization procedure not only reduces 50% spot welds of the double-hat welded crossbeam, but also relocates spot welds. Compared to the significant reduction of spot welds, the increase of displacement is acceptable.
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the example of torsional stiffness optimization of BIW in this paper. The optimization process is divided into two stages. At the first stage, a general analysis is performed to find out the critical regions that affect the performance significantly. After that, a local refined model with solid connecting elements between welding sheets is built and the solid connecting elements are defined as the topological optimization domain to give a local topological optimization for the structure. The other method is to apply this optimization method in conceptual design phrase of the BIW. It is feasible to use continuum solid material to represent the spot welds in most conceptual CAE models [18], which can be simplified by beams and shell elements. 5. Conclusions
Fig. 11. FE model of solid connection the welded sheets at the upper and lower D pillar of BIW.
This paper proposes a spot welds optimization procedure, where the welded sheets are connected by continuum solid elements instead of discrete elements. The process turns the discrete optimization problem into a continuous one. The connecting material is defined as design variable and topological optimization with the SIMP method is applied to remove the redundant material to obtain the minimum material volume which generates a layout for spot welds. The applications of spot welds optimization for a double-hat welded crossbeam and D pillar joints of a BIW show that the proposed method furnishes a good guideline for distributing the spot welds in spot-welded structures. The proposed spot welds optimization procedure could reduce the spot welds number effectively without noticeably sacrificing the performance of the spot-welded structure. The present study provides a feasible solution to spot-welded structures in promoting the welding quality and reducing cost. Acknowledgements
Fig. 12. The spot welds distributions are recommended by topology optimization result. The whole elements are modeled by continuum solid connecting elements to represent the original spot welds. With the topology optimization, the discrete red elements are recommended as the optimal spot welds for the structure, and the gray elements are redundant materials. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
An optimization is performed to a BIW, and a comparative study of torsional stiffness is conducted to verify the FE model of BIW. The analysis and rig test result are carried out under the same boundary conditions. The present study shows that the maximum error between the averaged test result and FEA is less than 10% in displacement at measuring points. The application to the BIW shows that the optimization method is feasible for complicated welded structures, although it is somewhat nontrivial to build a comprehensive model, such as a BIW, which is assembled with thousands of spots and welded sheets of complicated geometric shape. The present problem is how to efficiently apply the optimization method for the welded structure with a large number of elements or complex geometric structures. There are two potential methods to solve the problem. One of the methods is shown in
This work is supported by the National Natural Science Foundation of China (Grant No. 51475057), Fundamental Research Funds for the Central Universities (No. CDJZR14285501). X.J. would also like to acknowledge the supports from Chongqing City Science and Technology Program (No. cstc2013jcyjA0975). References [1] S.W. Chae, K.Y. Kwon, T.S. Lee, Finite Elem. Anal. Des. 38 (3) (2002) 227–294. [2] B. Bouyousfi, T. Sahraoui, S. Guessasma, K.T. Chaouch, Mater. Des. 28 (2) (2007) 414–419. [3] M. Pouranvari, H.R. Asgari, S.M. Mosavizadch, P.H. Marashi, M. Goodarzi, Sci. Technol. Weld. Joining 12 (3) (2007) 217–225. [4] A. Al-Samhan, S.M.H. Darwish, Int. J. Adhes. Adhes. 23 (1) (2003) 23–28. [5] S. Mahadevan, K. Ni, Reliab. Eng. Syst. Saf. 81 (1) (2003) 9–21. [6] S. Aslanlar, A. Ogur, U. Ozsarac, E. Ilhan, Mater. Des. 29 (7) (2008) 1427–1431. [7] P. Marashi, M. Pouranvari, S. Amirabdollahian, A. Abedi, M. Goodarzi, Mater. Sci. Eng.: A 480 (1) (2008) 175–180. [8] Y. Zhang, D. Taylor, Finite Elem. Anal. Des. 37 (12) (2001) 1113–1122. [9] Q.I. Bhatti, M. Ouisse, S. Cogan, Comput. Struct. 89 (17) (2011) 1697–1711. [10] A.H. Ertas, F.O. Sonmez, Finite Elem. Anal. Des. 47 (4) (2011) 413–423. [11] M.P. Bendsøe, N. Kikuchi, Comput. Meth. Appl. Mech. Eng. 71 (2) (1988) 197–224. [12] M. Bendsøe, Struct. Optim. 1 (4) (1989) 193–202. [13] M.R. Zhou, G.I.N. Rozvany, Comput. Meth. Appl. Mech. Eng. 89 (1) (1991) 309–336. [14] Y.M. Xie, G.P. Steven, Comput. Struct. 49 (5) (1993) 885–896. [15] R.T. Haftka, H.M. Adelman, Struct. Optim. 1 (3) (1989) 137–151. [16] R.T. Haftka, J.H. Starnes, AIAA J. 14 (6) (1976) 718–724. [17] L.A. Schmit, B. Farshi, AIAA J. 12 (5) (1974) 692–699. [18] M.A. Fard, T. Ishihara, H. Inooka, J. Biomech. Eng. 125 (4) (2003) 533–539.
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