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Robust Virtual Welding Process Optimization
ScienceDirect ScienceDirect Procedia Computer Science 00 (2018) 000–000
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Procedia Computer Science 00 (2018) 000–000
Procedia Computer Science 140 (2018) 342–350
Complex Adaptive Systems Conference with Theme: Cyber Physical Systems and Deep Learning, CAS 2018, 5 November – 7 November 2018, Chicago, Illinois, USA Complex Adaptive Systems Conference with Theme: Cyber Physical Systems and Deep Learning, CAS 2018, 5 November – 7 November 2018, Chicago, Illinois, USA
Robust Virtual Welding Process Optimization Robust Welding Process Vijay K Virtual Yalamanchili, Diego A Galindo,Optimization Justin C Mach Caterpillar, 510 Lake Cook Rd Suite 100, Deerfield, IL 60015. Vijay K Yalamanchili, Diego A Galindo, Justin C Mach Caterpillar, 510 Lake Cook Rd Suite 100, Deerfield, IL 60015.
Abstract The finished geometry of a welded structure is greatly affected by heat-induced distortion from the welding process. This often Abstract requires correction by straightening and top-level machining to ensure manufacturing quality. Distortion is sensitive to weld sequencing manufacturing environment. weld sequencing can significantly reduce distortion and manufacturing cost. The finishedand geometry of a welded structureOptimal is greatly affected by heat-induced distortion from the welding process. This often However, an optimalbyweld sequence and is nottop-level always intuitively for large structures with many welds. toVirtual requires correction straightening machining obvious, to ensureespecially manufacturing quality. Distortion is sensitive weld Fabrication and Technology (VFT)environment. is Caterpillar’s proprietary software for simulationreduce of welding physics via finite element sequencing manufacturing Optimal weld sequencing can the significantly distortion and manufacturing cost. analysis. an Powered the sequence highly successful VFTintuitively platform, obvious, a combinatorial genetic algorithm based However, optimalbyweld is not always especially for large structures withoptimization many welds.approach Virtual combined an uncertainty (UQ) module for evaluating robustness has been developed at Caterpillar weld Fabricationwith Technology (VFT)quantification is Caterpillar’s proprietary software for the simulation of welding physics via finite for element sequence optimization WSO has beenVFT successfully to more than 50 structures, resulting in the minimization of analysis. Powered by(WSO). the highly successful platform,applied a combinatorial genetic algorithm based optimization approach straightening The quantification combined VFT andmodule robustfor WSO approach results has in an accelerated process combined withoperations. an uncertainty (UQ) evaluating robustness been developedmanufacturing at Caterpillar for weld development schedule. (WSO). We will present a summary of our WSO approach and some results that demonstrate has proven sequence optimization WSO has been successfully applied to more than 50 structures, resulting inhow theWSO minimization of to be a transformative manufacturing planning at Caterpillar. straightening operations. The combined VFTtechnology and robust WSO approach results in an accelerated manufacturing process development schedule. We will present a summary of our WSO approach and some results that demonstrate how WSO has proven to be a transformative manufacturing planning technology at Caterpillar. © 2018 The Authors. Published by Elsevier B.V. © 2018 The Authors. by Elsevier B.V. This is an open accessPublished article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the Complex Adaptive Systems Conference with Theme: Engineering Cyber © 2018 The Authors. Published by Elsevier B.V. Selection and peer-review under responsibility of the Complex Adaptive Systems Conference with Theme: Engineering Cyber Physical This is anSystems. open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Physical Systems. Selection and peer-review under responsibility of the Complex Adaptive Systems Conference with Theme: Engineering Cyber Keywords: Virtual Manufacturing; Welding Simulation; Sequence Optimization; Genetic Algorithms; Uncertainity Quantification; Physical Systems. Keywords: Virtual Manufacturing; Welding Simulation; Sequence Optimization; Genetic Algorithms; Uncertainity Quantification;
1. Introduction
1. Introduction Controlling the distortion in a welded structure arising from the permanent deformation in the material due to the welding process is a critical challenge. Excessive distortion can cause many different problems, such as making it Controlling theadjacent distortion in a welded(as structure arising from the permanent deformation in the material due to the difficult to attach components in the pant-leg example later discussed) or deteriorating the functionality welding process is a critical Excessive can cause different problems, as making it of adjacent components (for challenge. instance, by causingdistortion uneven wear of themany friction pads of a motorsuch grader drawbar). difficult to attach adjacent components (as in the pant-leg example later discussed) or deteriorating the functionality of adjacent components instance, by causing uneven wear of the friction pads of a motor grader drawbar). 1877-0509 © 2018 The Authors.(for Published by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection peer-review under responsibility of the Complex 1877-0509and © 2018 The Authors. Published by Elsevier B.V. Adaptive Systems Conference with Theme: Engineering Cyber Physical Systems. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the Complex Adaptive Systems Conference with Theme: Engineering Cyber Physical Systems.
1877-0509 © 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the Complex Adaptive Systems Conference with Theme: Engineering Cyber Physical Systems. 10.1016/j.procs.2018.10.305
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Currently at Caterpillar, we have several simulation tools to predict the weld distortion given a set of process parameters. Process parameters such as weld sequence, inter pass cooling times and weld direction have significant impact on the resulting distortion and can be controlled relatively easily. Consequently, both experienced welding engineers and welding simulation analysts try to find the optimum process parameters that render a smaller distortion. However, this is in general no easy task, since a) finding an optimal sequence is a combinatorial problem, b) the problem itself is highly nonlinear, hence it is often difficult to obtain sequences with a small distortion in an intuitive manner, c) increasing cooling times does not always reduce the distortion, d) the simulation to predict the distortion is computationally expensive, typically on the order of hours, e) the behavior of the welded structure is further complicated by variability in environmental conditions, material properties, and variability in the welding process itself. Robotic welding reduces, but does not eliminate, this variability. All these challenges make it difficult to perform robust optimization of the welding sequence in an automated way. Researchers have been exploring optimizing the sequence of welding using simulation over the last two decades. Advances in manufacturing simulation and increasing availability of compute resources make solving this problem more feasible and lucrative. Most of the existing research surrounding this area revolves around optimizing the order of welding (or weld sequence). Huang et al [1], Segeborn et al [2], Kadivar et al [5], Vasudevan et al [7], Liao [6] have proposed variations of genetic algorithms to solve the problem, while Voutchkov et al [4], Asadi & Goldak [3] have presented a surrogate model approach. The aim of this work is to review welding process optimization and uncertainty quantification and demonstrate successful application to welding processes at Caterpillar. 2. Methods 2.1. Virtual Fabrication Technology In partnership with Battelle Memorial Institute, Caterpillar developed a suite of software tools in the late 1990s for simulating the physics of the arc welding process. Since then, VFT has been enhanced for usability and faster solving time. VFT can accurately predict the distortion and residual stresses that occur due to the welding process; it was developed specifically for simulating gas metal arc welding (GMAW) processes, but it can be adapted to model other arc welding processes as well. The distortion solutions that VFT provides for a specified process are what feed the process optimization algorithms that we discuss in this paper. To this end, a highly accurate thermal history is required to feed the de-coupled mechanical solution. The thermal history is typically provided by a proprietary thermal analytical solver that is part of the VFT package. This thermal history for the entire welding process is then provided as input to an FEA-based mechanical solver (Abaqus or WARP3D), which solves for the displacement and stresses throughout the entire welding process history. The mechanical solver uses a customized user material subroutine (UMAT) to represent the essential welding physics required to predict distortion. The deformation mechanisms that can be represented include deposition of filler material, annealing, hardening, and phase transformation. The computational time for welding process simulation is driven by the mesh size (like any other FEA) and the arc time of the process. Total computational (wall-clock) time can range from minutes to several days, depending on the model size and process arc time. 2.2 Welding Process Optimization As mentioned earlier, genetic algorithms (GAs) have been a popular approach for addressing such problems. The main difference between GAs used for non-combinatorial problems and those used for combinatorial problems lies in the chromosome used to represent each individual. In the former, the chromosome is a bit string, with each bit representing a value or level of each design variable; in the latter, the chromosome represents a sequence or combination. Simulations have revealed that inter-pass cooling times have nontrivial impact on the amount of distortion. Adding cooling times as design variables provides an opportunity to explore wider design spaces and thereby provide better solutions. However, this comes at the expense of a more complex optimization algorithm and longer run times. Since
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inter-pass cooling time is a non-combinatorial variable, we cannot extend the existing chromosome to accommodate the new variables. A new chromosome independent of the existing combinatorial chromosome (for welding sequence) is introduced to store inter-pass cooling times. This new chromosome has the same size as the combinatorial chromosome and each bit stores the cooling time after the corresponding weld sequence chromosome bit. The GA operators (crossover, mutation) operate on both the chromosomes while generating new individuals from parents. The crossover and mutation operators for combinatorial chromosome need to be tailored such that they reproduce a feasible sequence. Various crossover and mutation operators have been proposed and used for combinatorial chromosomes. We chose uniform crossover and order-changing mutation operators for the weld-sequence chromosome as depicted in Figure 1. Traditional uniform crossover and flip-bit mutation operators are applied to the cooling times chromosome. GA methods are in general better than gradient-based methods at attaining a global minimum; though, this is not guaranteed. We tested the proposed method on a benchmark problem with known solution, namely the blind traveling salesman problem. The standard traveling salesman problem consists of finding the sequence in which a salesman must visit several cities while minimizing total travel distance. The standard problem assumes upfront knowledge of the distance between cities. The blind traveling salesman formulation relaxes the assumption; the salesman can only compute the total travel distance after all the cities are visited. This problem resembles the problem at hand, since the resulting distortion is only known after all welding steps have been simulated. We tested the proposed method for a) a problem with eight cities and b) a problem with 51 cities, with known optimal solutions. Our method found the known optimal solution in case a., and a local optimum close to the known optimum in case b. (cf. Figure 2). Higher cooling times typically lead to lower distortion; hence the optimization algorithm might favor individuals with large cooling times between most welds. However, very high cycle times are not desirable from manufacturing perspective. Figure 3 depicts the Pareto front for the trade-off between the competing objectives of distortion and cycle time. To solve this multi-objective problem with GA’s, we incorporated Non-dominated Sorting Genetic Algorithm (NSGA-II) [9] approach with minor changes. The key advantages of NSGA-II over other approaches are a) reduced computational complexity, b) elitism, and c) diversity of individuals on the Pareto front (without having to specify additional parameters). For these reasons, NSGA-II has been extensively used for multi-objective optimization. The workflow for NSGA-II is summarized below (also see Figure 4): 1. Generate a random population for 'Generation 0' and evaluate individuals 2. Sort individuals to assign ranks and fitness to Gen 0 individuals 3. Select parent individuals based on fitness and perform GA operations (mutation, crossover) to generate population for 'Generation 1' similar to traditional GA 4. Create a union of '2N' individuals by merging the previous two generations and sort the union 5. Select 'N' individuals as potential parent individuals for the next generation based on sorting 6. Perform GA operations on 'N' parent individuals to obtain 'N' individuals for the next generation and evaluate individuals 7. If convergence has not been met and generation number is less than maximum number of generations, go back to step 3 Sorting operation for NSGA-II is based on two measures a) non-dominated sorting, b) crowding distance. Nondominated sort assigns ranks based on domination criteria. Individuals with one or more objective better than every other individual are assigned rank 1. Individuals that satisfy the domination criteria after rank 1 individuals are removed are assigned rank 2 and so on. Crowding distance sorts individuals with the same rank based on average distance of 2 points on either side of this individual along each objective. Individuals with more crowding distance are given priority to encourage diversity. Refer to [9] for a detailed explanation about NSGA-II. Selecting parents from parent pool for mating is done based on fitness using roulette wheel methodology. Fitness measure is assigned based on rank of the individual as shown below. Fi = 2(K - Ri) K is total number of ranks in the parent pool, and Ri is the rank of the individual. The implementation of multi-objective optimization is generic and can also be used to trade off distortion in different planes or other competing objectives specific to the problem.
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Many real-world welding operations on heavy structures require multi-pass welds at the same location. This means certain weld passes must be completed before others and incorporating these constraints upfront is critical to the optimization workflow for two reasons. 1) Weld simulations are computationally expensive and time consuming. Every infeasible solution explored is a waste of both time and money. 2) An infeasible solution with good fitness could derail the optimization and lead us in the direction of more infeasible solutions. The ability to impose constraints is an important piece of weld process optimization and facilitates feasible, more robust solutions. Below, we have incorporated constraints to aid the optimization algorithm. Subsequence constraints: These constraints allow the user to specify groups of welds that should be performed in a prescribed sequence. These constraints are enforced on each chromosome after GA operations (crossover and mutation). Rollover constraints: Many weld operations require repositioning of fixture, and it is desirable to limit the number of repositions (or rollovers) to manage cycle time in the absence of cycle time as an objective. Welds are grouped into different sides based on fixture position, and going from weld on one side to weld on another is counted as a rollover. These constraints are enforced after the sequence constraints on the chromosome. 2.3 Uncertainty Quantification Environmental, material, and some process parameters cannot be controlled as easily as weld sequence can, but variability in those parameters can have a significant effect on the welding performance. Uncertainty quantification analysis with respect to these variability sources can provide guidance on sequence selection for robust welding process. The aforementioned variability can be represented in parameterized simulation models by use of a probabilistic setting for the input parameters corresponding to those conditions. Given the probability distribution over the input parameters, the goal of the uncertainty quantification analysis is to sample from the solution map from the inputs to the distortions or other quantities of interest of the system. For expensive simulations such as those used for welding it is necessary to minimize the number of FEA solves needed. Many approaches have been proposed in the literature for propagation of uncertainty. These include spectral approximation methods such as the intrusive stochastic Galerkin methods that rely on distribution-adapted orthogonal polynomials, or non-instrusive stochastic collocation methods that take advantage of the smoothness frequently encountered in the quantities of interest to accelerate convergence of the UQ analysis [11]. Perturbation methods can also be used, such as in [12] which applies these methods to welding distortion uncertainty quantification and sensitivity analysis for various inputs. Here we restrict attention to three inputs to the simulation: speed, power, and cooling time. Distributional assumptions can be informed by collecting data from manufacturing facilities if available, or engineering judgement. A typical uncertainty quantification workflow consists of four stages: an initial design of experiments is used to generate training data for the surrogate model. The surrogate model is fit to that data. Then we propagate uncertainty from inputs to outputs by performing random sampling on the input parameters, estimating the resulting simulation outputs using the surrogate model, and finally summarize the resulting statistics in a way that is relevant for decision making. Decision makers may rely on summaries such as histograms of total distortion, summary statistics (mean distortion, distortion variance), or cost-weighted measures such as expected cost of straightening. The approach we have implemented uses an adaptive construction for the surrogate model, which requires iterations of adding to the DOE and improving the fit of the model. Approximation of the map from input parameters to distortion is achieved using Locally Adaptive Sparse Grid Stochastic Collocation as implemented by the open source TASMANIAN (Toolkit for Adaptive Stochastic Modeling and Non-Intrusive Approximation) package [13-15]. Stochastic Collocation methods are a class of function-approximation methods that offer fast convergence rates for a large class of functions, and are non-intrusive. The iterations of sample and model refinement are structured hierarchically, which ensures well-posed interpolation. Once the surrogate model has been constructed, it can be used to examine the sensitivity of the welding process with respect to speed, power, and cooling time. This can be used to select between candidate sequence solutions from the multi-objective optimization pareto curve, or to predict the variation in distortion that is to be expected in order to plan post-welding corrective actions, or to explore trade-offs between changes to process controls and distortion.
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3. Results In this section, we’ll review a few applications at Caterpillar that have benefited from weld process optimization. The first example is the “pant-leg” of a Cat® Wheel Loader (see Figure 6). This sub-assembly has 16 welds with tolerance requirements for flatness and leg opening. The baseline sequence satisfies the leg opening constraint but violates the flatness constraint by 285 percent. Using just the sequence optimization with rollover constraint, we satisfied both the leg-opening and flatness requirements with significant margins. In another application, introducing cooling times as design variables (see Figure 5) led to significant improvements. We achieved 25% lower distortion for the same cycle time, and a 50% reduction in cycle times for similar distortion when compared to fixed cooling times. For a T-joint weld (see Figure 7) two sequences were found to have similar amounts of distortion. Examining the robustness of these two candidate sequences demonstrated that sequence 1 had a narrower range of distortion outcomes, and sequence 2 had a significant right tail of larger distortion events. For a structure on Cat product X, the WSO analysis resulted in a 60%-70% reduction in distortion (see Figure 8). Uncertainty quantification (see Figure 9) permits further simulation-based planning, such as identifying opportunities to tighten process controls to meet specific distortion thresholds. 4. Conclusion Simulation tools like Virtual Fabrication Technology (VFT) model the complex physics of arc welding processes to predict distortion in fabricated structures. Weld sequence optimization based on a genetic algorithm and uncertainty quantification based on locally adaptive sparse grid stochastic collocation are combined with VFT to deliver robust optimization for welding that minimizes distortion and its variability. Several product examples showcase the power of these algorithms to create reductions that achieve quality targets and cost savings in manufacturing. 5. Illustrations
Fig. 1. Crossover and mutation operators.
Vijay K Yalamanchili et al. / Procedia Computer Science 140 (2018) 342–350 Yalamanchili et al./ Procedia Computer Science 00 (2018) 000–000
Fig. 2. Blind traveling salesman problem benchmark.
Fig. 3. Tradeoff between cycle time and distortion.
Fig. 4. NSGA-II Summary (Flowchart).
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Fig. 5. Effect of variable cooling times on distortion and cycle time.
Fig. 6. Weld process optimization (sequence only) effectiveness.
Fig. 7. y-distortion of T-joint weld a) sequence 1 and b) sequence 2.
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Fig. 8. Reduction in distortion along two axes due to weld sequence optimization.
Fig. 9. Uncertainty propagation illustrates the robustness of the candidate solutions.
Acknowledgements The authors would like to acknowledge Julian Norato, Sungmoon Jung, Joshua Webb, Arun Gain, Miroslav Stoyanov, and Badrinarayan Athreya for their contributions to the development of algorithms and tools used in this study.
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References [1] Huang, M., Hsieh, C. C. and Arora, J. S. (1997), A genetic algorithm for sequencing type problems in engineering design. Int. J. Numer. Meth. Engng., 40: 3105-3115. [2] Segeborn J, Carlson JS, Wärmefjord K, Söderberg R. Evaluating Genetic Algorithms on Welding Sequence Optimization With Respect to Dimensional Variation and Cycle Time. ASME. International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Volume 5: 37th Design Automation Conference, Parts A and B ():697-704. doi:10.1115/DETC2011-48393. [3] Asadi, M. & Goldak, J.A. Combinatorial optimization of weld sequence by using a surrogate model to mitigate a weld distortion. Int J Mech Mater Des (2011) 7: 123. [4] I. Voutchkov, A.J. Keane, A. Bhaskar, Tor M. Olsen, Weld sequence optimization: The use of surrogate models for solving sequential combinatorial problems, Computer Methods in Applied Mechanics and Engineering, Volume 194, Issues 30–33, 2005, Pages 3535-3551, ISSN 0045-7825. [5] Kadivar, M., Jafarpur, K. & Baradaran, G. Optimizing welding sequence with genetic algorithm. Computational Mechanics (2000) 26: 514. [6] Liao, Y. Optimal design of weld pattern in sheet metal assembly based on a genetic algorithm. Int J Adv Manuf Technol (2005) 26: 512. [7] M. Vasudevan, A. K. Bhaduri, Baldev Raj & K. Prasad Rao (2007) Genetic-Algorithm-Based Computational Models for Optimizing the Process Parameters of A-TIG Welding to Achieve Target Bead Geometry in Type 304 L(N) and 316 L(N) Stainless Steels, Materials and Manufacturing Processes, 22:5, 641-649. [8] Dhas, J & Jenkins Hexley Dhas, S. (2012). A Review on Optimization of Welding Process. Procedia Engineering. 38. 544–554. [9] Kalyanmoy Deb, Amrit Pratap, Sameer Agarwal and T. Meyarivan, "A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II", IEEE Transactions on evolutionary computation, Vol. 6, No. 2, April 2002. [10] Babuška, Ivo, Fabio Nobile, and Raúl Tempone (2007), “A stochastic collocation method for elliptic partial differential equations with random input data”, SIAM J. Numer. Anal., 45 (3): 1005-1034. [11] Bäck, Joakim, Fabio Nobile, Lorenzo Tamellini, and Raul Tempone (2011), “Stochastic spectral Galerkin and collocation methods for PDEs with random coefficients: A numerical comparison,” in Jan S. Hesthaven and Einar M. Rønquist (eds) Spectral and High Order Methods for Partial Differential Equations, Lecture Notes in Computational Science and Engineering 76, Springer, Berlin, Heidelberg [12] Michaleris, Panagiotis (2011), “Uncertainty quantification in modelling welding residual stress and distortion,” Science and Technology of Welding and Joining, 16 (8): 722-727. [13] Stoyanov, Miroslav K, and Clayton G. Webster (2016), “A dynamically adaptive sparse grids method for quasi-optimal interpolation of multidimensional functions,” Computers & Mathematics with Applications, 71 (11): 2449-2465. [14] Stoyanov, Miroslav K (2015), “User Manual: TASMANIAN Sparse Grids,” ORNL/TM-2015/596. [15] Stoyanov, Miroslav K, Damien Lebrun-Grandie, John Burkardt, and Drayton Munster (2013), https://github.com/ORNL/Tasmanian.
Available online at www.sciencedirect.com
Available online at www.sciencedirect.com
www.elsevier.com/locate/procedia
ScienceDirect
www.elsevier.com/locate/procedia
Procedia Computer Science 00 (2018) 000–000
Procedia Computer Science 140 (2018) 342–350
Complex Adaptive Systems Conference with Theme: Cyber Physical Systems and Deep Learning, CAS 2018, 5 November – 7 November 2018, Chicago, Illinois, USA Complex Adaptive Systems Conference with Theme: Cyber Physical Systems and Deep Learning, CAS 2018, 5 November – 7 November 2018, Chicago, Illinois, USA
Robust Virtual Welding Process Optimization Robust Welding Process Vijay K Virtual Yalamanchili, Diego A Galindo,Optimization Justin C Mach Caterpillar, 510 Lake Cook Rd Suite 100, Deerfield, IL 60015. Vijay K Yalamanchili, Diego A Galindo, Justin C Mach Caterpillar, 510 Lake Cook Rd Suite 100, Deerfield, IL 60015.
Abstract The finished geometry of a welded structure is greatly affected by heat-induced distortion from the welding process. This often Abstract requires correction by straightening and top-level machining to ensure manufacturing quality. Distortion is sensitive to weld sequencing manufacturing environment. weld sequencing can significantly reduce distortion and manufacturing cost. The finishedand geometry of a welded structureOptimal is greatly affected by heat-induced distortion from the welding process. This often However, an optimalbyweld sequence and is nottop-level always intuitively for large structures with many welds. toVirtual requires correction straightening machining obvious, to ensureespecially manufacturing quality. Distortion is sensitive weld Fabrication and Technology (VFT)environment. is Caterpillar’s proprietary software for simulationreduce of welding physics via finite element sequencing manufacturing Optimal weld sequencing can the significantly distortion and manufacturing cost. analysis. an Powered the sequence highly successful VFTintuitively platform, obvious, a combinatorial genetic algorithm based However, optimalbyweld is not always especially for large structures withoptimization many welds.approach Virtual combined an uncertainty (UQ) module for evaluating robustness has been developed at Caterpillar weld Fabricationwith Technology (VFT)quantification is Caterpillar’s proprietary software for the simulation of welding physics via finite for element sequence optimization WSO has beenVFT successfully to more than 50 structures, resulting in the minimization of analysis. Powered by(WSO). the highly successful platform,applied a combinatorial genetic algorithm based optimization approach straightening The quantification combined VFT andmodule robustfor WSO approach results has in an accelerated process combined withoperations. an uncertainty (UQ) evaluating robustness been developedmanufacturing at Caterpillar for weld development schedule. (WSO). We will present a summary of our WSO approach and some results that demonstrate has proven sequence optimization WSO has been successfully applied to more than 50 structures, resulting inhow theWSO minimization of to be a transformative manufacturing planning at Caterpillar. straightening operations. The combined VFTtechnology and robust WSO approach results in an accelerated manufacturing process development schedule. We will present a summary of our WSO approach and some results that demonstrate how WSO has proven to be a transformative manufacturing planning technology at Caterpillar. © 2018 The Authors. Published by Elsevier B.V. © 2018 The Authors. by Elsevier B.V. This is an open accessPublished article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the Complex Adaptive Systems Conference with Theme: Engineering Cyber © 2018 The Authors. Published by Elsevier B.V. Selection and peer-review under responsibility of the Complex Adaptive Systems Conference with Theme: Engineering Cyber Physical This is anSystems. open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Physical Systems. Selection and peer-review under responsibility of the Complex Adaptive Systems Conference with Theme: Engineering Cyber Keywords: Virtual Manufacturing; Welding Simulation; Sequence Optimization; Genetic Algorithms; Uncertainity Quantification; Physical Systems. Keywords: Virtual Manufacturing; Welding Simulation; Sequence Optimization; Genetic Algorithms; Uncertainity Quantification;
1. Introduction
1. Introduction Controlling the distortion in a welded structure arising from the permanent deformation in the material due to the welding process is a critical challenge. Excessive distortion can cause many different problems, such as making it Controlling theadjacent distortion in a welded(as structure arising from the permanent deformation in the material due to the difficult to attach components in the pant-leg example later discussed) or deteriorating the functionality welding process is a critical Excessive can cause different problems, as making it of adjacent components (for challenge. instance, by causingdistortion uneven wear of themany friction pads of a motorsuch grader drawbar). difficult to attach adjacent components (as in the pant-leg example later discussed) or deteriorating the functionality of adjacent components instance, by causing uneven wear of the friction pads of a motor grader drawbar). 1877-0509 © 2018 The Authors.(for Published by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection peer-review under responsibility of the Complex 1877-0509and © 2018 The Authors. Published by Elsevier B.V. Adaptive Systems Conference with Theme: Engineering Cyber Physical Systems. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the Complex Adaptive Systems Conference with Theme: Engineering Cyber Physical Systems.
1877-0509 © 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the Complex Adaptive Systems Conference with Theme: Engineering Cyber Physical Systems. 10.1016/j.procs.2018.10.305
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Currently at Caterpillar, we have several simulation tools to predict the weld distortion given a set of process parameters. Process parameters such as weld sequence, inter pass cooling times and weld direction have significant impact on the resulting distortion and can be controlled relatively easily. Consequently, both experienced welding engineers and welding simulation analysts try to find the optimum process parameters that render a smaller distortion. However, this is in general no easy task, since a) finding an optimal sequence is a combinatorial problem, b) the problem itself is highly nonlinear, hence it is often difficult to obtain sequences with a small distortion in an intuitive manner, c) increasing cooling times does not always reduce the distortion, d) the simulation to predict the distortion is computationally expensive, typically on the order of hours, e) the behavior of the welded structure is further complicated by variability in environmental conditions, material properties, and variability in the welding process itself. Robotic welding reduces, but does not eliminate, this variability. All these challenges make it difficult to perform robust optimization of the welding sequence in an automated way. Researchers have been exploring optimizing the sequence of welding using simulation over the last two decades. Advances in manufacturing simulation and increasing availability of compute resources make solving this problem more feasible and lucrative. Most of the existing research surrounding this area revolves around optimizing the order of welding (or weld sequence). Huang et al [1], Segeborn et al [2], Kadivar et al [5], Vasudevan et al [7], Liao [6] have proposed variations of genetic algorithms to solve the problem, while Voutchkov et al [4], Asadi & Goldak [3] have presented a surrogate model approach. The aim of this work is to review welding process optimization and uncertainty quantification and demonstrate successful application to welding processes at Caterpillar. 2. Methods 2.1. Virtual Fabrication Technology In partnership with Battelle Memorial Institute, Caterpillar developed a suite of software tools in the late 1990s for simulating the physics of the arc welding process. Since then, VFT has been enhanced for usability and faster solving time. VFT can accurately predict the distortion and residual stresses that occur due to the welding process; it was developed specifically for simulating gas metal arc welding (GMAW) processes, but it can be adapted to model other arc welding processes as well. The distortion solutions that VFT provides for a specified process are what feed the process optimization algorithms that we discuss in this paper. To this end, a highly accurate thermal history is required to feed the de-coupled mechanical solution. The thermal history is typically provided by a proprietary thermal analytical solver that is part of the VFT package. This thermal history for the entire welding process is then provided as input to an FEA-based mechanical solver (Abaqus or WARP3D), which solves for the displacement and stresses throughout the entire welding process history. The mechanical solver uses a customized user material subroutine (UMAT) to represent the essential welding physics required to predict distortion. The deformation mechanisms that can be represented include deposition of filler material, annealing, hardening, and phase transformation. The computational time for welding process simulation is driven by the mesh size (like any other FEA) and the arc time of the process. Total computational (wall-clock) time can range from minutes to several days, depending on the model size and process arc time. 2.2 Welding Process Optimization As mentioned earlier, genetic algorithms (GAs) have been a popular approach for addressing such problems. The main difference between GAs used for non-combinatorial problems and those used for combinatorial problems lies in the chromosome used to represent each individual. In the former, the chromosome is a bit string, with each bit representing a value or level of each design variable; in the latter, the chromosome represents a sequence or combination. Simulations have revealed that inter-pass cooling times have nontrivial impact on the amount of distortion. Adding cooling times as design variables provides an opportunity to explore wider design spaces and thereby provide better solutions. However, this comes at the expense of a more complex optimization algorithm and longer run times. Since
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inter-pass cooling time is a non-combinatorial variable, we cannot extend the existing chromosome to accommodate the new variables. A new chromosome independent of the existing combinatorial chromosome (for welding sequence) is introduced to store inter-pass cooling times. This new chromosome has the same size as the combinatorial chromosome and each bit stores the cooling time after the corresponding weld sequence chromosome bit. The GA operators (crossover, mutation) operate on both the chromosomes while generating new individuals from parents. The crossover and mutation operators for combinatorial chromosome need to be tailored such that they reproduce a feasible sequence. Various crossover and mutation operators have been proposed and used for combinatorial chromosomes. We chose uniform crossover and order-changing mutation operators for the weld-sequence chromosome as depicted in Figure 1. Traditional uniform crossover and flip-bit mutation operators are applied to the cooling times chromosome. GA methods are in general better than gradient-based methods at attaining a global minimum; though, this is not guaranteed. We tested the proposed method on a benchmark problem with known solution, namely the blind traveling salesman problem. The standard traveling salesman problem consists of finding the sequence in which a salesman must visit several cities while minimizing total travel distance. The standard problem assumes upfront knowledge of the distance between cities. The blind traveling salesman formulation relaxes the assumption; the salesman can only compute the total travel distance after all the cities are visited. This problem resembles the problem at hand, since the resulting distortion is only known after all welding steps have been simulated. We tested the proposed method for a) a problem with eight cities and b) a problem with 51 cities, with known optimal solutions. Our method found the known optimal solution in case a., and a local optimum close to the known optimum in case b. (cf. Figure 2). Higher cooling times typically lead to lower distortion; hence the optimization algorithm might favor individuals with large cooling times between most welds. However, very high cycle times are not desirable from manufacturing perspective. Figure 3 depicts the Pareto front for the trade-off between the competing objectives of distortion and cycle time. To solve this multi-objective problem with GA’s, we incorporated Non-dominated Sorting Genetic Algorithm (NSGA-II) [9] approach with minor changes. The key advantages of NSGA-II over other approaches are a) reduced computational complexity, b) elitism, and c) diversity of individuals on the Pareto front (without having to specify additional parameters). For these reasons, NSGA-II has been extensively used for multi-objective optimization. The workflow for NSGA-II is summarized below (also see Figure 4): 1. Generate a random population for 'Generation 0' and evaluate individuals 2. Sort individuals to assign ranks and fitness to Gen 0 individuals 3. Select parent individuals based on fitness and perform GA operations (mutation, crossover) to generate population for 'Generation 1' similar to traditional GA 4. Create a union of '2N' individuals by merging the previous two generations and sort the union 5. Select 'N' individuals as potential parent individuals for the next generation based on sorting 6. Perform GA operations on 'N' parent individuals to obtain 'N' individuals for the next generation and evaluate individuals 7. If convergence has not been met and generation number is less than maximum number of generations, go back to step 3 Sorting operation for NSGA-II is based on two measures a) non-dominated sorting, b) crowding distance. Nondominated sort assigns ranks based on domination criteria. Individuals with one or more objective better than every other individual are assigned rank 1. Individuals that satisfy the domination criteria after rank 1 individuals are removed are assigned rank 2 and so on. Crowding distance sorts individuals with the same rank based on average distance of 2 points on either side of this individual along each objective. Individuals with more crowding distance are given priority to encourage diversity. Refer to [9] for a detailed explanation about NSGA-II. Selecting parents from parent pool for mating is done based on fitness using roulette wheel methodology. Fitness measure is assigned based on rank of the individual as shown below. Fi = 2(K - Ri) K is total number of ranks in the parent pool, and Ri is the rank of the individual. The implementation of multi-objective optimization is generic and can also be used to trade off distortion in different planes or other competing objectives specific to the problem.
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Many real-world welding operations on heavy structures require multi-pass welds at the same location. This means certain weld passes must be completed before others and incorporating these constraints upfront is critical to the optimization workflow for two reasons. 1) Weld simulations are computationally expensive and time consuming. Every infeasible solution explored is a waste of both time and money. 2) An infeasible solution with good fitness could derail the optimization and lead us in the direction of more infeasible solutions. The ability to impose constraints is an important piece of weld process optimization and facilitates feasible, more robust solutions. Below, we have incorporated constraints to aid the optimization algorithm. Subsequence constraints: These constraints allow the user to specify groups of welds that should be performed in a prescribed sequence. These constraints are enforced on each chromosome after GA operations (crossover and mutation). Rollover constraints: Many weld operations require repositioning of fixture, and it is desirable to limit the number of repositions (or rollovers) to manage cycle time in the absence of cycle time as an objective. Welds are grouped into different sides based on fixture position, and going from weld on one side to weld on another is counted as a rollover. These constraints are enforced after the sequence constraints on the chromosome. 2.3 Uncertainty Quantification Environmental, material, and some process parameters cannot be controlled as easily as weld sequence can, but variability in those parameters can have a significant effect on the welding performance. Uncertainty quantification analysis with respect to these variability sources can provide guidance on sequence selection for robust welding process. The aforementioned variability can be represented in parameterized simulation models by use of a probabilistic setting for the input parameters corresponding to those conditions. Given the probability distribution over the input parameters, the goal of the uncertainty quantification analysis is to sample from the solution map from the inputs to the distortions or other quantities of interest of the system. For expensive simulations such as those used for welding it is necessary to minimize the number of FEA solves needed. Many approaches have been proposed in the literature for propagation of uncertainty. These include spectral approximation methods such as the intrusive stochastic Galerkin methods that rely on distribution-adapted orthogonal polynomials, or non-instrusive stochastic collocation methods that take advantage of the smoothness frequently encountered in the quantities of interest to accelerate convergence of the UQ analysis [11]. Perturbation methods can also be used, such as in [12] which applies these methods to welding distortion uncertainty quantification and sensitivity analysis for various inputs. Here we restrict attention to three inputs to the simulation: speed, power, and cooling time. Distributional assumptions can be informed by collecting data from manufacturing facilities if available, or engineering judgement. A typical uncertainty quantification workflow consists of four stages: an initial design of experiments is used to generate training data for the surrogate model. The surrogate model is fit to that data. Then we propagate uncertainty from inputs to outputs by performing random sampling on the input parameters, estimating the resulting simulation outputs using the surrogate model, and finally summarize the resulting statistics in a way that is relevant for decision making. Decision makers may rely on summaries such as histograms of total distortion, summary statistics (mean distortion, distortion variance), or cost-weighted measures such as expected cost of straightening. The approach we have implemented uses an adaptive construction for the surrogate model, which requires iterations of adding to the DOE and improving the fit of the model. Approximation of the map from input parameters to distortion is achieved using Locally Adaptive Sparse Grid Stochastic Collocation as implemented by the open source TASMANIAN (Toolkit for Adaptive Stochastic Modeling and Non-Intrusive Approximation) package [13-15]. Stochastic Collocation methods are a class of function-approximation methods that offer fast convergence rates for a large class of functions, and are non-intrusive. The iterations of sample and model refinement are structured hierarchically, which ensures well-posed interpolation. Once the surrogate model has been constructed, it can be used to examine the sensitivity of the welding process with respect to speed, power, and cooling time. This can be used to select between candidate sequence solutions from the multi-objective optimization pareto curve, or to predict the variation in distortion that is to be expected in order to plan post-welding corrective actions, or to explore trade-offs between changes to process controls and distortion.
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3. Results In this section, we’ll review a few applications at Caterpillar that have benefited from weld process optimization. The first example is the “pant-leg” of a Cat® Wheel Loader (see Figure 6). This sub-assembly has 16 welds with tolerance requirements for flatness and leg opening. The baseline sequence satisfies the leg opening constraint but violates the flatness constraint by 285 percent. Using just the sequence optimization with rollover constraint, we satisfied both the leg-opening and flatness requirements with significant margins. In another application, introducing cooling times as design variables (see Figure 5) led to significant improvements. We achieved 25% lower distortion for the same cycle time, and a 50% reduction in cycle times for similar distortion when compared to fixed cooling times. For a T-joint weld (see Figure 7) two sequences were found to have similar amounts of distortion. Examining the robustness of these two candidate sequences demonstrated that sequence 1 had a narrower range of distortion outcomes, and sequence 2 had a significant right tail of larger distortion events. For a structure on Cat product X, the WSO analysis resulted in a 60%-70% reduction in distortion (see Figure 8). Uncertainty quantification (see Figure 9) permits further simulation-based planning, such as identifying opportunities to tighten process controls to meet specific distortion thresholds. 4. Conclusion Simulation tools like Virtual Fabrication Technology (VFT) model the complex physics of arc welding processes to predict distortion in fabricated structures. Weld sequence optimization based on a genetic algorithm and uncertainty quantification based on locally adaptive sparse grid stochastic collocation are combined with VFT to deliver robust optimization for welding that minimizes distortion and its variability. Several product examples showcase the power of these algorithms to create reductions that achieve quality targets and cost savings in manufacturing. 5. Illustrations
Fig. 1. Crossover and mutation operators.
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Fig. 2. Blind traveling salesman problem benchmark.
Fig. 3. Tradeoff between cycle time and distortion.
Fig. 4. NSGA-II Summary (Flowchart).
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Fig. 5. Effect of variable cooling times on distortion and cycle time.
Fig. 6. Weld process optimization (sequence only) effectiveness.
Fig. 7. y-distortion of T-joint weld a) sequence 1 and b) sequence 2.
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Fig. 8. Reduction in distortion along two axes due to weld sequence optimization.
Fig. 9. Uncertainty propagation illustrates the robustness of the candidate solutions.
Acknowledgements The authors would like to acknowledge Julian Norato, Sungmoon Jung, Joshua Webb, Arun Gain, Miroslav Stoyanov, and Badrinarayan Athreya for their contributions to the development of algorithms and tools used in this study.
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