Procedia CIRP 12 ( 2013 ) 25 – 30

Available online at www.sciencedirect.com

2212-8271 © 2013 The Authors. Published by Elsevier B.V.Selection and peer review under responsibility of Professor Roberto Tetidoi: 10.1016/j.procir.2013.09.006

ScienceDirect

8th CIRP Conference on Intelligent Computation in Manufacturing Engineering

Incremental generation of hierarchical meshes for the thermomechanical simulation of NC-milling processes

R. Jolieta,*, A. Byfutb, T. Surmanna, A. Schröderb aTU Dortmund, Institute of Machining Technology, Baroper Str. 301, 44227 Dortmund, Germany

bHumboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany * Corresponding author. Tel.: +49-231-755-2364; fax: +49-231-755-5141.E-mail address:joliet@isf.de.

Abstract

For the production of large structural components, the NC-milling process is often run with minimum quantity lubrication only. Additionally, most of the raw material will be removed by using process parameters that lead to a significant generation of heat. To guarantee error-free products, the thermally induced deformation has to be taken into account. Thus, a simulation of the milling process, which calculates the thermal expansion, can be used to predict deviations and to optimize milling strategies. In this paper a new approach for the simulation of the NC-milling process is presented. It combines a pure geometrical simulation with new techniques of adaptive finite element methods. The key for a fast simulation of linear thermo elasticity is a hierarchical mesh, based on hexahedra with additional geometric primitives on the lowest level of refinement. Using hexahedra permits the application of tensor-product approaches to achieve faster computations, whereas the use of constrained-approximation allows for multi-level hanging nodes in the mesh. In turn, hp-adaptive finite element methods can be applied. The investigation focuses on the basic coupling of the simulation components and the incremental generation of suitable meshes for the finite element analysis and thereby completely avoids any remeshing action. It is based on a hierarchical octree that is extended by an adaptive sub-mesh for each cell that is partially removed during the simulated cutting process. Thus, the mesh is incrementally refined and adapted during the simulation. Based on previous experimental and analytic investigations, a complete simulation cycle will be presented. For the prediction of the heat input, an empirical model has been developed in order to ensure a fast simulation. After the remote finite element analysis, local displacements are applied to the position of the milling tool instantly. In this way very accurate results can be achieved that serve as a basis for further process optimization operations. © 2012 The Authors. Published by Elsevier B.V. Selection and/or peer-review under responsibility of Professor Roberto Teti.

Keywords: Deformation; Finite element method (FEM); Milling; Modelling; Simulation; Thermal error.

1. Introduction

The application of minimum quantity lubrication in the roughing operation of the NC-milling process can result in a significant heat input into the workpiece. This applies also for aluminum materials that are typical for large structural components. The resulting deformation due to thermo-elastic effects can be significant and affects the finishing operation too. To guarantee error-free products, a simulation of the milling process, which calculates the thermal expansion, can be used to predict deviations and to optimize milling strategies.

In this paper, a new approach for the simulation of the

NC-milling process is presented that uses a hybrid simulation system. It combines a geometric simulation of the material removal with finite element analysis (FEA) to compute distortions. The key for a fast simulation of linear thermo elasticity is a hierarchical mesh, based on hexahedra. Additionally, geometric primitives are used on the lowest level of refinement of the mesh to produce a very accurate approximation of the surface. Using hexahedra permits the application of tensor-product approaches to achieve a faster computation, whereas the use of constrained-approximation allows for multi-level hanging nodes in the mesh. In turn, hp-adaptive finite element methods can be applied.

© 2013 The Authors. Published by Elsevier B.V.Selection and peer review under responsibility of Professor Roberto Teti

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In the following, all relevant aspects of the model of the workpiece and the generation of a mesh for the FEA are introduced. Next, the concept of the hybrid simulation system is presented.

2. Heat Input During NC-Milling

During chip formation some fraction of the cutting work that is applied by the machine is converted to heat and is inserted into the workpiece. The exact percentage of the heat input depends on the constantly changing tool engagement and boundary conditions like material properties. Earlier experiments have shown how to measure these boundary conditions of the milling process and a way to predict the resulting heat input by fast simulations has been developed [1]. The basic idea is to compute the chipping form in order to determine the cutting force along each cutting edge for its sweep through the material. This makes it possible to calculate the cutting work by assuming a constant cutting force during each time step. The time-discrete rotation results in well-defined sampling positions on the chipping form where the force is calculated.

Using this technique, time-dependent boundary conditions for finite element methods can be computed. Next, material removal and the generation of a mesh for the FEA are described.

3. Milling Simulation

The pure geometric simulation of a NC-milling operation is commercially available in several CAM-systems. Those cover machining times, material removal, or the prediction of contact conditions of the milling tool. Additionally, collision detection algorithms can be applied and are often used before parts are assembled on a machine. Current research topics cover the simulation of tool, machine, or workpiece vibrations [2, 3] and can be used to predict and avoid negative influences of chatter on the quality of a workpiece or on the lifetime of machines and tools.

In the following, modeling techniques and implementation details, which are important for the simulation of thermomechanical deformations, are presented and discussed.

3.1. Computation of cutting force and cutting work

Apart from simulations using finite element analysis, many simulative approaches of the milling process, which take the dynamic of the tool into account, make use of adequate models to predict process forces. A well-known model for this is the Kienzle equation, which calculates the cutting forces for any given chipping thickness based on experimentally determined

parameters. The method to transfer this model for the process of turning to the milling process has been described already in detail in [4]. Roughly, the exact theoretic shape of each chip is computed using the Constructive Solid Geometry (CSG) and the chip is sampled to determine the chipping thickness piecewise. Next, the cutting work can be calculated and a model for the heat input makes time-dependent simulations of the workpiece during milling possible [1].

3.2. Model of the workpiece

For the computation of the chipping form, a local model of the workpiece is used, which is created by combining primitive solids such as spheres, cylinders, or tori with Boolean operators [5]. An adaptation of this technique makes it possible to create an exact model of the entire workpiece [6]. Both models can be used to simulate dynamic aspects of the milling process and to predict surface structures accurately. However, for a representation of the workpiece with finite elements these models are neither well-structured nor easy to process.

On the other hand, the challenge of manipulating meshes directly by finding intersections and inclusions for every edge of each cell is a computationally intensive task. Based on the complexity of a regular mesh that increases with a power of three when accuracy is raised, the amount of operations that must be performed to update a mesh can consume significant runtime of a simulation. Additionally, various quality criteria have to be considered for the generated mesh to guarantee low computational errors and fast convergence rates during the solution of the underlying mathematical problem.

For a fast simulation of linear thermo elasticity, a hierarchical mesh is proposed, which is based on nested hexahedra with additional geometric primitives on the lowest level of refinement. Using hexahedra permits the application of tensor-product approaches to speedup computation, whereas the use of constrained-approximation allows for multi-level hanging nodes in the mesh. Combined, hp-adaptive finite element methods can be applied [7].

3.3. Material removal and mesh generation

Given an initial model of the workpiece that is represented by one hexahedron, a previously developed simulation of the 5-axis milling process is performed. For each timestep, the position and the orientation of the tool are calculated and the given CSG model can be used to perform the simulation of material removal. Each hexahedron, which intersects the model of the tool, is recursively subdivided into 8 sub cells until a predefined maximum level of refinement is reached. The removal of

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completely covered cells leads to a typically scattered surface on which no accurate boundary of the workpieceis defined. Therefore, the octree is enriched on thelowest level with further geometric primitives such as prisms, pyramids, and tetrahedra. This structure is called a graph tree which describes an enriched octree in general. Other information, such as additional points of the surface, can be added as well to improve visualdebugging or the accuracy of the representation. Thisguarantees a good approximation of the resulting surfaceand allows for high quality FEA results.

As can be seen clearly from Figures 1 and 2, the refinement strategy creates new subdivisions when needed and in every step a valid mesh is availablewithout any remeshing.

Fig. 1. Hierarchical structure of nestedhexahedra and approximation of the surface

Fig. 2. Visualization of the basic refinement scheme and removal of cells that have been covered completely by the model of the tool.

On the one hand, each cell has to be independent toavoid effects of the subdivision of a cell on itsneighbors. On the other hand, shared corners, edges, andfaces of adjacent hexahedra can be used to speed up theupdate mechanism during the simulation of materialremoval.

To profit from the alignment of hexahedra, collinearedges of cells should be intersected only once with themodel of the tool. Thus, a hidden model that representsshared edges is necessary and must not exceed runtimeor memory complexity of the actual octree model.

The number of cells in the octree does not increase by a factor of 8 per level of refinement. In fact, an additional step of refinement increases the resulting

number of cells by a factor from approximately 2 to 6,depending on the level and on the actual contour that isproduced. For more accurate refinements, the scalingfactor is about 4.

Figure 3 shows a verification run for the givenimplementation that displays the number of cells for an increasing maximum level of refinement. Abrupt changes in the gradient of the plot are caused by thestandard roughing strategy with coplanar layers to bemachined.

Fig. 3. Number of cells during the simulation of material removal withdifferent maximum refinement levels.

A corresponding mesh is shown in Figure 2.For the implementation of the hidden model that is

used for updating the hierarchical mesh, dexel boards have been added to the model of the workpiece [8]. Each dexel defines multiple pairs of entry and exit pointsalong a common line. Thus, it represents a spatialdiscretization of a volumetric domain. Aligning an array of dexels in a grid structure, a dexel board is defined that is capable to sample two spatial dimensions and the third dimension is represented with floating point precision.

In the presented approach, each dexel is aligned alongthe edges of the hexahedra. By intersecting each dexelwith the model of the tool, the affected hexahedra can befound easily and each cell of the sub mesh can beupdated.

Figure 4 shows a sequence of steps that are performedduring the updating process of the mesh. For each cell,the simulation keeps track whether its corners have been removed by the model of the tool.

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Fig. 4. (a) Hidden dexel boards and shape of tool (gray); (b) dexel chip; (c) intersection points and status of corners; (d) resulting mesh.

For edges with differently marked corners, one adittional intersection point is stored to allow for precise sub meshing decisions. Similar to the marching cubes algorithm, a sub-mesh pattern is selected by analyzing the state of the corners. Then, geometric primitives are selected to fill the volume.

The selection of geometric primitives to represent the non-removed parts of a hexahedron, depending on the status of its edges, is crucial for the resulting mesh. To avoid other hanging nodes and edges, except those of nested hexahedra, a subdivision scheme for the faces of subdivided hexahedron has been selected. Figure 5 shows all possible configurations of corners for one face.

Fig. 5. Configurations for the subdivision of faces of a hexahedron.

By using only these subdivisions each cell can be analyzed independently and no promotion of the resulting structure to neighbors is needed. On the downside there exist some configurations for hexahedra that cannot be filled with the given primitives by only adding new diagonals into the cube. The resolution of this problem can be found by adding new nodes inside the hexahedron. The simplest example is the cube that is cut into two pieces horizontally with the upper corners removed and the lower corners not yet intersected by the model of the tool. Without the assumption that all four vertical intersections are coplanar, the resulting cube-shaped part has to be represented by pyramids since each face must be rectangular and the bottom of a pyramid. The conclusion of this configuration must then be a composition of pyramids that share a common inner node at their apexes. A corresponding example can be found in Figure 6.

Fig. 6. Configurations for different states of a hexahedron and an example of surface approximation.

Fig. 7. Intersection points of hidden dexel model and surface approximation with tetrahedra (dark blue), pyramids (light blue), and prisms (green).

The presented algorithm for the incremental generation of meshes is a prerequisite for the hybrid simulation approach that will be described in the following. It meets the described criteria for hp-adaptive finite element analysis and can generate meshes for FEA very fast. Finally, a new application of existing material removal techniques has been presented that allows for a direct simulation of the cutting process to an existing mesh for FEA.

4. Hybrid Simulation System

Based on the simulation of the milling process, a mesh is created that represents the workpiece. Starting with a solid hexahedron, elements are subdivided and removed incrementally while the model of the tool moves through the workpiece.

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Additionally, information about heat input is calculated for each timestep and located on the mesh in the contact zone. This can be calculated using the method described in section 3.1. All this is transferred over a network interface to a remote server that is capable of performing the actual hp-adaptive finite element analysis. For this, a model of linear elasticity is used that gives a temperature distribution for each timestep as well as the deformation of the mesh resulting from thermal elongation. At this point, the most important step is to transfer this information back to the process simulation to guarantee the next step to be calculated accurately. In this way, the varying, local deformation of the mesh during production can be taken into account. In turn, the mesh for FEA is manipulated directly in an online simulation.

The precise material removal is performed by inverse transformation of the model of the tool. This guarantees an accurate, valid, and fast generation of meshes during the simulation cycle. Together, thermally induced deformations during milling can be simulated and errors that are caused by this effect can be predicted.

Figure 8 presents results from a first test of the hybrid simulation approach. Subfigure (b) displays the variable polynomial degrees for cells of different size. Figures (c) and (d) show first results of a simulation run. They were produced for testing purposes only and have not been compared with real experiments yet. The validation of the whole system is pending but first comparisons gave promising results.

Fig. 8. (a) Process simulation, material removal and meshing; (b) polynomial degree; (c) temperature; (d) von Mises stress

5. Outlook

A hybrid simulation system has been presented that is capable of taking deformations of the workpiece during milling into account. Due to by thermo-elastic material behavior, no precomputation is possible and a direct and exact material removal on the FE mesh is a challenging task. The presented method makes it possible to generate meshes for the described scenario incrementally. Thus, any remeshing during simulation will be avoided and yet a well-conditioned and accurate mesh is generated in each step of the simulation. The quality of meshes, the accuracy of the finite element methods, and the performance of the simulation have to be evaluated profoundly. After this, further optimizations can be applied to the meshing scheme and to the hp-adaptive finite element analysis. Different techniques to reduce the overall number of cell in the mesh by a different representation of the surface of the workpiece are to be investigated. Furthermore, adaptive refinement schemes and alternative heat input model can be examined to improve the accuracy of results.

Acknowledgements

This paper is based on investigations of the project

CutSim) which is kindly supported by the German Research Foundation (DFG).

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