Optimal riser design in sand casting process with evolutionary topology optimization

Transcription

1 Struct Multidisc Optim (2009) 38: DOI /s z INDUSTRIAL APPLICATION Optimal design in sand casting process with evolutionary topology optimization R. Tavakoli P. Davami Received: 23 April 2007 / Revised: 6 May 2008 / Accepted: 8 June 2008 / Published online: 17 July 2008 Springer-Verlag 2008 Abstract The optimal design of a casting feeding system is considered. The topology is systematically modified to minimize the volume, while simultaneously ensuring that no defect appears in the product. In this approach, we combine finite-difference analysis of the solidification process with evolutionary topology optimization to systematically improve the feeding system design. The outstanding features of the presented method are: its efficiency, ease of implementation and simplifying definition of the initial design. The efficiency and capability of the presented method are supported by illustrative examples. Keywords Casting optimization Feeder design Riser optimization 1 Introduction Casting is an important manufacturing process for a variety of industries. Among different casting processes, sand casting is the most widely used process for both ferrous and non-ferrous metals. Solidification of the molten metal after being poured into a mold cavity is an important phase in the casting process which greatly affects on product quality and yield. To make up shrinkage during the phase change, the freezing fronts take R. Tavakoli (B) P. Davami Department of Material Science and Engineering, Sharif University of Technology, P.O. Box , Tehran, Iran tav@mehr.sharif.edu the required liquid from adjacent liquid regions. The last freezing regions are the most probable locations of shrinkage cavities. Risers are appended to the casting at suitable locations to provide directional solidification from the casting to s such that the shrinkage defects are placed within the s. The total volume of s should be minimized to improve the casting yield and productivity (Ravi and Srinivasan 1996). During the two past decades, the casting simulation has been widely used in foundry industry to evaluate designs and to predict formation of the shrinkage cavities. Nowadays, the available casting simulation softwares accept a user s design and then analyze the design to predict the likelihood of defects. Once an analysis has been completed, the results should be analyzed by the user, and if an area of potential defect is found within the casting (such as internal shrinkage porosity), then it is needed to make some logical modifications in the design and repeat the simulation until the desired result is obtained. Thus the traditional trial-and-error design cycle on the foundry floor has been replaced with trial-and-error on the computer. In recent years, a number of papers have been published which report successful use of numerical optimization methods to optimize design (Morthland et al. 1995; Lewis et al. 2001). In these works, the optimal design is formulated as a shape optimization problem which is solved by the gradient based minimization methods. The objective function to be minimized is defined as the volume and the constraints are defined such that the directional solidification along a priori defined feeding path is provided. The main disadvantages of the works mentioned above are: high computational cost due to direct sensitivity analysis, the need for a nearly feasible initial de-

2 206 R. Tavakoli, P. Davami sign and definition of feeding path. To overcome these difficulties, in Tavakoli and Davami (2008) the topology optimization based on SIMP approach is adapted to find the best topology for a prescribed value of the volume. In Tavakoli and Davami (2007c), a heuristic growth method is suggested to automate the optimal design in the sand casting process. This work is in fact in parallel with the present study as it could be considered as an additive evolutionary topology optimization method (considering the present work which is a subtractive evolutionary algorithm). In the present study, the optimal design in the sand casting process is formulated as an evolutionary topology optimization problem (cf. Xie and Steven 1997). The initial design of the presented method is a casting with an over-designed (s). In fact this method needs the (s) layout, i.e., the number of s and their relative configuration, as an initial design. Then, the optimization method gradually improves the design by minimizing the total volume of (s), simultaneously ensuring a defect-free casting. 2 Evolutionary optimization algorithm The evolutionary topology optimization (ETO) method firstly developed by Xie and Steven (1993) to optimize topology of structures subject to static loads. Later, Li et al. (1999) extended application of ETO to optimize shape and topology of steady-state 2D heat conduction problems. Although there is no theoretical support for optimality of ESO methods, they work fine in practice and usually generate competitive results in contrast to the gradient based topology optimization methods. In this section, we adapt the ETO method for optimal design in the sand casting process. The initial design of ETO method is an overestimated design, i.e., the initial design is healthy from functionality viewpoints (in our application, initial design generates a defect-free casting) but it is not efficient from consumed material resource points of view (casting yield 1 in our application). Denoting the spatial domain by, we called the mentioned overestimated design as the feasible design domain, design. Note that the final (optimal) design will be a subset of design. Prior to start of analysis, should be decomposed into sufficiently 1 casting yield = casting weight (without ) / weight of consumed molten metal fine cubic elements. In practice, these elements are same as the computational spatial grid which is used to solve governing equations. In the present study, each element of the discretized spatial domain is called a voxel. Every voxel has a topology indicator flag, w, where w = 1 inside the main phase and w = 0 inside the background phase. In our application the main phase is the molten metal and the background phase is the sand mold. Therefore w = 1 inside the casting, w = 0 inside the sand mold and w takes either 0 or 1 inside design (varies during topology evolution). At the starting point of optimization, w = 1 inside design and we have an overestimated design. The objective of ETO in optimal design is to find the best distribution of w inside design, that produces a sound casting with minimum consumption of the molten metal. For this purpose, low performance voxels (w = 1) are gradually removed from the by altering the corresponding topology indicator flag to zero (w 0). Note that in this manner, at every stage of optimization we have a sharp 0 1 topology which is suitable for real world applications. Since the must be separated from the casting after complete solidification, the connection area of the -casting which is called as the -neck, should be minimized too. For this purpose, we separate the feasible -neck design domain, design neck,form design and optimize each of the and -neck separately in their corresponding feasible design domains. 2.1 Initial design As mentioned in the above section, the initial design of the presented method is a feasible, over designed (s) topology. For this purpose, the casting should be placed in a suitable mold box and then its layout should be defined. The layout contains information about number of s and their approximate positions and shapes. The conventional design rules (e.g. see: Carlson et al. 2002; Ouetal.2002; Campbell 2004) could be used to define the layout. A simple definition of the layout could be some boxes which are intersected with the casting (see parts G of Figs. 2, 3, 4 and 5). In this manner, the user considers each to feed a portion of casting which is called as the region under feed (RUF) of the corresponding in this study. In our algorithm s do not have direct connection to each others (connected s automatically considered as one ), but RUFs could overlap with each others. In the initial design stage user can exclude infeasible regions from the design domains ( design and design neck ).

3 Optimal design in sand casting process 207 For example connection of s to the bottom and high curvature surfaces should be avoided as much as possible. 2.2 Evolutionary design optimization Providing the initial design, an evolutionary algorithm is performed iteratively until some stopping criteria are satisfied. The details of procedure is as follows. At every iteration, the energy equation is solved (see: Appendix 1 for more details) and the performance indexes of s s voxels are computed. Since the s should be solidified later than the casting (to provide the directional solidification), the performance index (for every melt voxel) is defined as the local freezing time which is computed during solution of the energy equation. Computing the performance index field, the following operation is performed for each separately. The melt voxels (voxels with w = 1) ofl, R l,are sorted based on an ascending order of the performance index. Then, the first n l voxels of R l are converted to the sand voxels (w 0), where n l = RRr l Vr l, RRr l is the rejection ratio of R l (input parameter) which indicates volume percent of R l that should be removed during each evolution iteration. Vl r is the volume at the current iteration. 2.3 Evolutionary -neck design optimization Design of -neck is performed in the -neck feasible design domain. design neck is composed from the first n shell shells of the mold voxels started from the casting surface which are members of the initial layout. n shell is computed based on the -neck length which is a user defined parameter. The -neck voxels are removed iteratively in the same manner as described for the voxels. Since the performance of the is strictly a function of its corresponding neck, an additional criterion should be used during -neck evaluation to avoid -neck freezing prior to solidification of its corresponding RUF. For this purpose, the critical -neck contact area, A cr = α A r, is defined for each -neck and the evaluation of a -neck is suppressed when its contact area with either the casting or be smaller than A cr,wherea r is the current area of and 0 <α<1is a user defined parameter. 2.4 Stopping criteria It is clear that the discussed removal procedure gradually reduces the size until it completely disappears. Therefore the removal procedure should be terminated based on some stopping criteria. In the present study we use two types of stopping criteria: global and local. The global criterion is a user defined parameter, e.g., maximum casting yield. It is obvious that the casting yield is increased gradually during evolution iterations. The local stopping criteria (constraints) are defined for every and if a local constraint is satisfied for a R l, the removal procedure is suppressed for R l and continued for other s, or the last optimization cycle is repeated with a smaller rejection ratio. In this study we use the maximum allowable macro shrinkage appearance in the RUF of each as local constraints. These upper bounds are some user defined parameters that are expressed as the defect percent which is computed by the following relation 100 (volume of defect in RUF)/(volume of RUF). This measure is computed by the macro-shrinkage prediction algorithm (for more details see: Appendix 2). If the removal procedure is stopped for all s or the global stopping criterion is met, the optimization is discarded and the last design is taken as the optimal design. 2.5 Overall algorithm The algorithm used in the present study is summarized in this section. 1. Initialization: definition of the initial layout ( design and design neck ) and control parameters. 2. Solidification analysis and defect prediction (see: Appendixes 1 and 2). 3. Riser design modification: for each l that does not meet the local stopping criteria, do: 3.1. Compute volume, Vl r Compute n l = RRr l Vr l Sort the voxels in an ascending order of the local freezing time Convert the first n l voxels of the sorted list to the sand voxels. 4. Riser-neck design modification: for each - l that does not meet the local stopping criteria, do: 4.1. Compute -neck volume, Vl n Compute n l neck = RRn l Vn l 4.3. Sort the -neck voxels in an ascending order of the local freezing time Convert the first n l neck voxels of the sorted list to the sand voxels. 5. If the global stopping criteria is not satisfied then Goto step 2.

4 208 R. Tavakoli, P. Davami Table 1 Physical properties, initial and boundary conditions used in our simulations Casting Sand mold Density (Kg/m 3 ) 7,300 (solid) 1,500 6,826 (liquid) Heat capacity (J/Kg C) 627 1,128.6 Heat conductivity (W/m C) Latent heat (J/Kg) Initial temperature ( C) 1, T liq ( C) 1,488 T sol ( C) 1,439 Interfacial heat transfer coefficient (W/m 2 C) Casting Mold Mold Air Results and discussion In this section we present demonstrative examples to show potential of the presented method for optimal design in the sand casting process. For this purpose casting of a low alloy carbon steel in a silica sand is considered. The physical properties and initial and boundary conditions used in our analysis are presented in Table 1. The applied optimization parameters are as follows. The formation of macro-shrinkage (maximum 0.5 %) in RUF of each is considered as our local stopping criterion. The rejection ratio (RR) was equal to 0.2 in the first four iterations (iterations 1 4), 0.15 in the second four iterations (iterations 5 8), in the third four iterations (iterations 9 12) and 0.05 in the remained iterations for all s and -necks. α =0.1 and n shell = 4 for all -necks. Four test cases (see Fig. 1) have been studied are to evaluate the capability of the presented method. After the voxelization procedure, the total number of voxels and number of metal voxels (in the initial design) are (392,000, 199,381), (1,057,160, 397,195), (1,705,280, 781,062) and (272,734, 1,197,727) for test cases 1 4 respectively. As the computing platform we have used a personal computer with an AMD 2.41 GHz CPU and 2GB RAM. The opensource software package Cart- Gen (Tavakoli 2007) has been used for grid generation purpose in this study. Figures 2, 3, 4 and 5 show result of topology optimization related to test cases 1 4 respectively. Sections A F of each plot show the variation of (s) topology, macro-shrinkage and contour plot of the local freezing time (in second) during optimization procedure. Sections G and H of each plot show the initial layout and the final optimized topology respectively. The casting yield correspond to the initial and the final (optimal) designs are: (14.9%, 72.2%), (20.6%, 73.5%), (11.6%, 62.5%) and (10.6%, 62.9%) for test cases 1 through 4 respectively. The total CPU time, CPU time of the first and the last optimization steps are (in minutes): (13.3, 2.3, 0.35), (8.5, 1.15, 0.16), (38.3, 5.72, 0.56) and (94.5, 19.1, 0.81) for test cases 1 4 respectively. The results show that the directional solidification is preserved during optimization while the casting yield is increased gradually and the final design is free from the macro-shrinkage defects (of course based on the prescribed threshold). Unlike other topology optimization methods that usually suffer from a high computational cost, that of the presented method seems to be tolerable. Fig. 1 Configuration of test cases used in the present study: test-case 1 (a), test-case 2 (b), test-case 3 (c), test-case 4 (d) a b c d

5 Optimal design in sand casting process 209 A B C D E F Fig. 2 Result of test case 1: the variation of topology, macroshrinkage and contour plot of local freezing time (in second) during optimization, A F are related to iterations: 1, 4, 8, 16, 20 G H and 24 respectively, the initial layout G vs. final optimized topology H

6 210 R. Tavakoli, P. Davami A B C D E F G H Fig. 3 Result of test case 2: the variation of topology, macro-shrinkage and contour plot of local freezing time (in second) during optimization, A F are related to iterations: 1, 4, 8, 12, 16 and 18 respectively, the initial layout G vs. final topology H

7 Optimal design in sand casting process 211 A B C D E F G H Fig. 4 Result of test case 3: the variation of topology, macro-shrinkage and contour plot of local freezing time (in second) during optimization, A F are related to iterations: 1, 4, 8, 16, 20 and 24 respectively, the initial layout G vs. final topology H

8 212 R. Tavakoli, P. Davami A B C D E F G H Fig. 5 Result of test case 4: the variation of topology, macro-shrinkage and contour plot of local freezing time (in second) during optimization, A F are related to iterations: 1, 4, 8, 16, 20 and 24 respectively, the initial layout G vs. final topology H

9 Optimal design in sand casting process Conclusions The evolutionary topology optimization method is adapted for optimal design in the sand casting process. The method starts with an over-estimated design (in fact a layout) and gradually improves the casting yield, simultaneously ensures producing a defect-free casting. The efficiency, ease of implementation and promoting a good degree of automation is the outstanding features of the presented method in contrast to the other alternative methods. This method could be appended to the traditional commercial casting simulation packages with a little effort. Demonstrative examples show efficiency and success of the presented method for optimal design in the sand casting process. Appendix 1: solidification analysis Assume the spatial domain is denoted by = c m, where c is a portion of which includes the casting and m denotes the mold region. The contact surface of casting and mold is denoted by Ɣ c m = c m.from a macroscopic point of view, if the effect of melt flow during solidification is neglected, the solidification is governed by the heat conduction equation as follows (cf. Lewis and Ravindran 2000): condition in the mold region is θ m =θ in m {t =0}, where θ is the ambient temperature. The natural boundary condition is applied at the mold-air interface, i.e., k θ m n m a = h (θ θ m ) on Ɣ m a [0, T], whereh denotes the mold-air convective heat transfer coefficient, Ɣ m a is the mold-air interface and n m a is the unit normal vector directed toward the environment which is defined on Ɣ m a. Following (Manzari et al. 2000), the cast-mold interface is modeled by the interfacial θ heat transfer coefficient method, i.e., k c θ c n i = k m m n i = h i (θ c θ m ) on Ɣ c m [0, T],wheren i is the unit normal vector directed toward the mold which is defined on Ɣ c m and h i is the local heat transfer coefficient of castmold interface. The interfacial heat transfer coefficient could be determined by experiment, direct numerical simulation or combination of experiment and numerical simulation (cf. Lewis and Ransing 1998). Since (1) and (2) should be solved within every topology evolution iteration, efficient numerical solution of these equations is a key to develop a practical optimization tool. In Tavakoli and Davami (2007a) a fast solidification solver has been introduced to solve conduction dominated solidification problems. Later, authors (Tavakoli and Davami 2007b) extend application of this solver to simulate real world sand casting process. This solver has been used in the present study for the purpose of solidification analysis. For more details about accuracy and efficiently of this method, refer to: (Tavakoli and Davami 2007b). θ c ρ c c c t = (κ c θ c )+ρ c L f s t in Q c = c [0, T] (1) where ρ c is the cast density, c c is the cast specific heat, θ c is the temperature field in the casting region, t is the time variable, κ c is the cast thermal conductivity, L is the fusion latent heat, f s is the solid fraction field, and [0, T] is the temporal domain. The initial condition is θ c = θ 0 c in c {t = 0}, where θ 0 c is the pouring temperature. In the mold region we have: ρ m c m θ m t = (κ m θ m ) in Q m = m [0, T] (2) where ρ m is the mold density, c m is the mold specific heat, θ m is the temperature field in the mold region and κ m is the mold thermal conductivity. The initial Appendix 2: solidification defects prediction As stated in Section 2.4, predicting occurrence of solidification defects in the RUF of each is essential prior to each evolution cycle. There are several types of numerical models that can be employed to predict defects formation in metals due to solidification shrinkage (for an state-of-the-art review see: Stefanescu 2005). In the present study, the simplified shrinkage model suggested by Imafuku and Chijiiwa (1983) is applied to predict formation of solidification defects in the casting. Of course other models could be employed without any limitation. The main benefit of the simplified shrinkage model is its high efficiency besides the reasonable accuracy (for more details see: Imafuku and Chijiiwa 1983).

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