In this paper, both software model visualization with path simulation and associated machining product are produced based on the step ring-based three-axis path planning to demo model-driven graphics processing unit (GPU) feature in tool path planning and 3D image model classification by GPU simulation. Subtractive 3D printing (i.e., 3D machining) is represented as integration between 3D printing modeling and computer numerical control (CNC) machining via GPU simulated software. Path planning is applied through visualization of surface material removal in high-resolution and 3D path simulation via ring selective path planning based on accessibility of path through pattern selection. First, the step ring selects critical features to reconstruct computer-aided design (CAD) design model as stereolithography (STL) voxel, and then, local optimization is attained within interested ring area for time and energy saving of GPU volume generation as compared to global automatic path planning with longer latency. The reconstructed CAD model comes from an original sample (GATech buzz) with 2D image information. CAD model for optimization and validation is adopted to sustain manufacturing reproduction based on system simulation feedback. To avoid collision with the produced path from retraction path, we pick adaptive ring path generation and prediction in each planning iteration, which may also minimize material removal. Moreover, we did partition analysis and G-code optimization for large-scale model and high density volume data. Image classification and grid analysis based on adaptive 3D tree depth are proposed for multilevel set partition of the model to define no cutting zones. After that, accessibility map is computed based on accessibility space for rotational angular space of path orientation to compare step ring-based pass planning verses global path planning of all geometries. Feature analysis via central processing unit (CPU) or GPU processor for GPU map computation contributes to high-performance computing and cloud computing potential through parallel computing application of subtractive 3D printing in the future.

## Introduction

Subtractive 3D printing using CNC machine is a new way of 3D printing with machining capability and diverse material flexibility. Conventional 3D printing works through a 3D printer with additive path planning to build successive layers while the subtractive 3D printing simulates material removal running on CNC machine with software running via GPU parallel computing features. The motivation is to have quality material shape generated without air pollution and better control through computer program via automated CNC machine with G-code and less time in path planning and model validation by simulation. Due to busy schedule and higher hourly rate of machine shop, we aim at operating with the right G-code of a model for less machining time within reasonable amount of time given cutting-edge capability of CNC to make hard material part with strong cutting force. For simple software demo of model-drive path planning in three-axis case, we pick material simulation for 3D isoscallop visualization in software path planning and machining so that real-time visualization can be displayed in graphics for dynamic simulation of material removal on the model surface during machining via adaptive cutting depth and step distance. We pick step-ring selective path planning to demonstrate our results.

Isoscallop analysis is a cared topic in machining operation as the isoscallop scale and shape affect the surface quality of machining parts, and has thus been studied by tons of research literature. We also inspect the fact of isoscallop effects but in software perspective of how three-axis path planning step distance may affect machining output of surface geometry and quality. Since the software simulation highly maps the real machining output after particular tool sizing experiment, the simulation in 3D volume geometry in each pass can demo the real machining output of actual mapping tool in full develop mode. The 3D volume visualization of cutting operation happens by 3D geometry rendering of each material removal process that actually runs on GPU to reduce the parallel latency between user interaction and software rendering. Optimal planning can be attained if path planning time is estimated in a better way.

The major research gap is in model-based 3D visualization by path parameter tuning via GPU simulation and the modeling of system integration for ring selective prediction without path retraction. By predictive data analysis through feedback, we would have optimized modeling of 3D CAD design and computer-aided manufacturing (CAM) path planning with more tool control in embedded system software; this is one of the motivations and contributions. Another research gap is parallel data analysis of accessibility map computation [1] through adaptive depth-based volume rendering by path on GPU when we consider orientation. We compute accessibility map generation in a data analytical way through accessible map sequence formulating 3D accessible space, such as branch volume effects by varying path planning depth and picking CPU or GPU as a processor. One of the research gaps rests in picking image designed CAD model and parameter tuning in STL file format (such as additive manufacturing) to run three-axis CAM path planning algorithm with good machining sensitivity.

In this paper, as software and machining test demo, we use adaptive ring selection and feedback control logic to estimate each ring path and get adaptive G-code partition, and have three-axis machining path mapping the GPU simulation via model-driven user design.

## Previous Work and Related Techniques

Model validation played an important role in manufacturing in subtractive 3D printing, not only for effective production but also for sustainability of entire system. For sustainable manufacturing, model validation procedure is proposed in system point of view during dynamics for efficiency of labor productivity, with sensitivity and behavior pattern usually included during model validation [2].

On the other hand, for parallel computation capability development and service-oriented model, cloud manufacturing and service-oriented model is also studied as public and hybrid platform as well as private community platform for key challenges in manufacturing to merge with new technologies in cloud computing [3]. Parallel computing in software thus becomes a front side pioneer of these new technologies for promoting the scalable and distributed computing design or feasibility study for these platform-dependent customer services. CUDA application, famous in parallel computing and high-performance computing, has called the attention of software developer for graphic and visualization application in all fields including manufacturing application design for more user friendly integration.

The aim of flexible manufacturing systems (FMSs) is to achieve efficiency for mass production, and piecewise linear approximations and heuristic algorithms are reviewed to work for nonlinear constraints-based objective functions [4]. Local path planning in linearization and nonlinear constraints thus may contribute to mass production for the nonlinear objective-based model-oriented design so that a perfectly balanced system can be attained. To understand the relation of flexible parameter selection and get a processor application point of view between local path planning and global path planning based on selective model, and especially model with complicated geometry and structure, we provide data analysis into software methodologies based on graphic analysis and software application; this is also why we apply predictive data analysis with GPU simulation. Our motivation is to demo an image-based product design and its feasibility in local path planning by ring selection based on patterns of model.

Regarding the geometry of model and parameter validation, the voxel model is good for detailed structural analysis and topology design study of surface modeling and optimization not only for automation industry but also for computer-aided design in software [5]. Manufacturing grid system and its optimal selection-based analysis in resource and service have been studied [6–10]. Computational application in software graphic grid and structure may also improve the model-based application study for automatic parameter setting and provide the mapping solutions for simulation.

In fact, the voxel model techniques are highly applied in graphics-based data application and simulation. Study in Ref. [11] gives an idea of how to generate lookup table, block density, as well as index buffer for 2D and 3D textures via GPU with simple voxel and marching cube introduction. Hamada [12] introduced indirect fast multipole method-boundary element method for voxel model analysis and applied field for electrical field of biological samples via GPU acceleration and pseudocode template for CUDA kernels of the fast multipole method, along with multiple graphics processing unit execution. Crassin [13] introduced a system that achieves real-time rendering performance for several billion voxels and how the introduced prefiltered geometry model and approximate cone tracing method can achieve blurry effects and real-time indirect lighting. Rasterization and scheduling of fragment shading is also attached at the end. He also proposed a solution based on an adaptive data representation depending on the current view and occlusion information, coupled to an efficient ray-casting rendering algorithm, extracting information during rendering to guide data production and streaming [14]. The resulting solution can efficiently render large volumetric datasets interactively to real-time rendering performance for several billion voxels. Rees and McColgan [15] provided statistical analysis for multiple comparison and general methods of preprocessing as well as fluid registration for voxel-based morphometry.

Simulations based on grid and voxel analysis are also very popular in details of model regions and fluid dynamics. Sugiyama et al. [16] proposed a new simulation method for solving fluid–structure coupling problem, which has been developed with all the basic equations numerically solved on a fixed Cartesian grid for fluid and solid motions using a finite difference scheme with various verifications and validations of the present full Eulerian method with numerical accuracy. Scahill et al. [17] described a voxel-based analysis of nonlinear-registered serial magnetic resonance (MR) imaging to demonstrate the most statistically significant regions and global entropy in advancing of Alzheimer’s disease (AD). Differences between groups by statistical analysis and change detection of individual overtime by fluid registration are introduced after magnetic resonance (MR) imaging acquisition and preprocessing. Study in Ref. [18] introduced issues and methods such as steady-state temperature initialization involved in accurately predict specific absorption rate (SAR). The anatomical complexity and high degree of spatial resolution are required to capture the predicted SAR “hot spots” via the development of the realistic anatomical datasets, in which the voxel-based descriptions are created by dividing the space occupied by the object being described (e.g., the human body) into a 3D grid of small, equal sized volume elements known as voxels. In Ref. [19], a new simulation method for solving fluid–structure coupling problems suitable for voxel-based geometry has been developed, and a volume-of-fluid approach is applied to describing the multicomponent geometry. Study in Ref. [20] introduced finite element analysis as an invaluable tool for investigating the biomechanical function of complex skeletal structures. Alternative method is also introduced using a voxel-based finite element mesh where each computerized tomography voxel is converted directly into a finite element, with no loss of detail. The voxel-based finite element analysis on millions of voxels/elements that are needed to model small sections, as well as issues such as color-mapping strains, clipping, and adding displacement field, is discussed in Ref. [21]. Regarding software algorithms in material surface machining, software algorithms based on the product surface isoscallop analysis are presented with strategies based on product scallop heights and parametric equations of sculpture surface quality measurement [22].

In this paper, we apply voxel model in 3D printing software to demo accessible visualization of 3D ring step based on path planning via GPU computation so that CUDA application in high-performance computing and high-precision manufacturing can be improved. Based on model selective parameter tuning and user-oriented design, we are motivated to attain optimal depth and step in path planning for subtractive 3D printing. In addition, we aim to study a flexible planning of local path planning based on ring selective path versus global path planning using accessibility map via different processors so that future mass production is feasible for flexible manufacturing system and sustainable manufacturing with efficiency even for large-scale model and high density data. With optimal selection and freedom of path selection by user choices in GPU software platform, flexibility of manufacturing system can be enhanced in path planning with multi-axis CNC machining control capability to promote processor accelerated access and data efficiency in storage. In addition, we provide dynamic depth-based 3D grid analysis based on image classification on selected buzz model for GPU path planning topology. For future improvement, we aim to develop minimization and system workload balance with optimal division based on model-based structure and user-oriented modeling design by parallel computing.

## System Problems and Model-Driven Big Data

In our project, we pick data structure mainly using hybrid dynamic tree (HDT) [23] and 3D contact volume rendering for sparse geometry representation as key aspect of our GPU parallel path planning algorithm and simulation foundation of subtractive 3D printing.

The STL model gets into mesh format and is then loaded into hybrid dynamic tree data structure shown in Fig. 1 with selected model topology data based on initial input of 3D volume generation. Hybrid dynamic tree has been formulated from static data structure with enhanced dynamic data structure for each of its cubic division into subtree structures for model geometry element representation. From the data structure of HDT topology, we can get simple branch modeling within a certain scale in the tree as

The above equation displays the relation of branches in an HDT that formulates data structure for topology data searching and workload processing. We will study the branch relation with accessibility map and volume generation in the Results of Simulation and Accessibility Map section.

The original map can be 3D bitmap and the data are dynamically generated to fit the sizing at distributed locations via different tree branches so that each tree element can represent analytical information of a model that gets loaded via the tree structure. Usually, the model after tree representation has been divided into three HDT states: inside, boundary, and outside. The part that gets outside of a tree representation structure can either be marked as empty or off-boundary. To define the surface quality of the model and branch data consolidation, the intersection of tree and model with half-fulfilled state of tree leaf element is considered as boundary. The inner side of tree is usually marked as full state in its tree element representation. The HDT gets most of its operation through GPU and 3D rendering speed, and thus gets enhanced speed up a lot compared to CPU version as the normal tree representation over CPU incurs heavy workload even in parallel program with full workload containment for a single pc memory and load capability to sustain. The HDT data structure gets depth formulation based on model-driven workload and would have timing difference due to its topology and tree dimension as data searching or rendering has to go through the tree depth level in order to reach its target. Assume the platform is fixed with given topology of model, large workload due to complex geometry shape or higher component number of computational geometry will have longer processing time. Thus, the data structure is related to the model-driven workload with platform latency and traverse time of data loading.

The entire model-driven system can be viewed from Fig. 2 as illustration of how the system data flow works through the optimal path planning based on modeling and geometry analysis. The problem can then be viewed as the input of image into a series of complex operations toward optimal path planning and analysis. In this paper, we focus on the three-axis path planning without the rotational movement of machining platform in milling mode. Thus, we have to pick selective model that can be made in three-axis machining path setting that the CNC device will take in its operation range. Figure 2 describes the system flow after the model input for economic analysis based on planning and how the modeling information may help with the early phase of planning for model path selection based on the multi-axis range of CNC platform and whether time and cost are accessible based on a particular model.

With sculptprint software as key path planning testbed, we have to face further problems in our machining operation of subtractive 3D printing. The listed problems are as follows:

data efficiency and load limit within operation range and safe zone

simulation accuracy and data flow in loop tuning based on system feedback of sustainable process

machine sensitivity based on model geometry

theoretical and systematic view in software application and data structure for model and ring path prediction

parallel platform and GPU computation feasibility study via local path planning based on partial geometry pattern compared to global path planning of full geometry model

The most important part is model parameter tuning and STL remodeling, so that we could set the right parameters of model to fit software rendering requirement of model and satisfy enough machining sensitivity while planning the path with optimal orientation in 3D space.

## Ring Path Planning and System Scope

Local versus global path planning in data comparison is displayed in Fig. 3, which annotates the concepts of step ring-based path planning as local selection within boundary verses global path planning in all data generation of full voxel models.

What we present as step ring-based path planning is local optimization based on geometry with more user freedom to select the patterns and path parameters based on the model geometry for machining. Thus, the surface complexity of the ring selective models can be higher than other simple geometry models. Local path planning is thus more focused on selected data within interest region rather than global data generation of full model. Ring step is selective based on regions of ring set cover on target geometry patterns. The contribution is minimal material removal with most surface pattern geometry maintained without generating too much nonlinear conditions for path retraction while machining the model and simulating path visualization by GPU.

Compared to similar work in 3D printing and tool path planning, ring pass selection is similar to machining spiral strategy [24] but differs by GPU simulation for path planning of selected ring set cover of model in software visualization. The number of ring distribution and ring cover set is selected based on the step over distance of path patterns users prefer. In contrast, machining spiral strategy only means the path of machining in trajectory form of tool travel in space, and it is subject to the spiral distance of machining so that the part surface is smooth enough to attain the machining quality, although it is not that convenient to control the machining spiral gap of tool travel due to the seamless tool movement along its moving direction.

## Parameter Tuning by System Feedback

The system loop for simulation operation can be described in Fig. 4 as we pick two-way simulation movement for feedback: First, we pick simulation in software to give prefiltered error analysis if an early path planning error or retraction is generated to avoid collision by path inspection. Then, for the machining operation based on software simulation (given the simulation software in accurate form would be cost dependent and highly on accuracy of actual machine setup and measurement input), we pick easy software simulation regardless of actual machine setting; thus, the preselected path planning parameters may still be inaccurate for further machining test, and we also pick the machining process feedback based on the system loop to give further feedback for parameter tuning based on actual planning path.

Figure 4 describes the double feedback system loop with simulation while we pick model and path editing parameters for optimal model geometry and size based on machine range availability and axis layout so that enough sensitivity and path planning feature may be displayed to our interest.

## Model Validation for Sustainability

The model validation process is similar to Fig. 4, which is actually how we finally confirm a successful path planning operation based on right model dimension and orientation.

Figure 5 displays the model validation process which shows many loops for feedback of simulation on a particular model; even though a geometry pattern is selected, we can still adjust the sizing and input of the model via the 3D software modeling tools. Although we can pick the geometry model by any design interest, we still need to respect the limit of machining capability while we do path planning. Based on the machine we use for the CNC machining operation, we could have options such as three-axis, four-axis, and five-axis machines. The machine formatting and axis setting do affect the path planning output as we have to consider whether the model geometry has enough freedom with orientation rotation choices while we do the path planning.

The CNC platform is more like *x*, *y*, and *z* axis layout for three-axis machine which gives capability of linear path planning capability along all space but the machine tool can only face the model geometry horizontal plane in perpendicular form which limits the rotation within its own area of no oriented freedom while following the path planning. The buzz model could be machined by the five-axis machine with both of its faces operated on, as the five-axis machine is assumed to work toward any complexity of geometry that is available to operate in space. But we first try the three-axis machine to see if the reduced complexity model path planning can be made based on a complex patterned model coming from CAD and original image logo design. Thus, the buzz model is more like a test path planning run though the automatic path planning of whole body geometry path planning is considered more software friendly and user ease to operate on without any selection of process parameter data. Our aim is to do user manual tuning of parameters based on simulation of 3D path along the model and volume rendering of actual geometry surface after machining based on our proposed path planning parameters. Thus, we have more freedom in choosing the pending time of path planning at each of its operation phase. If we pick depth of path planning deeper at a time we would have less iteration of path planning to cut toward the target depth we expect for material removal, but we would also have longer waiting time while generating all paths during the volume rendering for global path planning.

The step ring-based path planning applied in three-axis path considers the geometry orientation before actual whole surface is covered for automatic path generation in software. Thus, we can select based on preknowledge of a model if we have enough close inspection of the model topology or level set in space to test whether we can generate minimal material removal based on path planning to optimize the stock geometry and orientation position in machining. Complex geometry analysis with multidistribution of height layout involves the model validation process and works for sensitivity and path correction in adaptive modeling in CAD. For example, the buzz model has each pair set of the buzz antenna and wing on different height planes of surface.

## Ring Solution and Simulation Methods

### Image Reconstruction Into CAD Model and Model Parameter Tuning for G-Code Path Planning.

The pattern represented geometry can be both symmetric and made easily from conventional CAD software and nonsymmetric model that comes from complex images for model contour shape data. The images of buzz figure are extracted for information with its shape contour key points, and these points then help formulate the 3D geometry model as input to the software platform for path planning and G-code generation. The images can be loaded into CAD software tools such as CATIA for further production of CAD design and parameter tuning. We pick the 3D buzz model for simulation-based feedback control on its model sizing and dimensions in 3dmax software to get enough sensitivity and geometry shaping based on simulation output of G-code to produce model adaptable machining via model geometry parameters, which should satisfy right software rendering simulation, enough shape completeness, and machine sensitivity for noticeable curves in machining within physical feasibility of path planning. The sensitivity of machining motions of CNC tool path based on a model should demonstrate enough cutting depth to satisfy product quality; this is why we aim for model optimization for enough sensitivity.

### Pattern Selective Path Planning on Local Geometry and Optimization by Simulation.

Ring selective path planning works based on local geometry and partial selection of model region that has interesting features to focus, and the volume computation time compared to full path rendering by volume computation is much less via GPU based on saving of full model path computation time. Thus, the selection can be solely on the local geometry pattern we care to produce with local optimization rather to cover entire model for all path generation and global optimization.

### The Predicted Ring Step Estimation Based on Simulation.

The ring is picked to maximize geometry cover in surface without details across lines causing retraction and collision. The retraction condition can be viewed as constraints that keep collision avoidance of model-based path planning. Automatic path planning considers filter based on retraction so that the path is based on orientation that is three-axis with vertical tool tip toward the stock piece. Linear retraction is generated when a gap between two points is too large for a tool tip to travel. We pick initial ring pass to cover the model pattern and additively increase the ring number based on the depth geometry given the ring does not violate the retraction rule and generate no retraction path causing collision. As long as a retraction in path does not happen, we make linear ring path planning by increasing the ring number, and we reduce the ring number and the acceleration of ring number by following the iteration of path planning while there is a retraction based on the simulation feedback or the machining feedback.

### Adaptive Step-Ring Selection in Three-Axis Path Planning.

How we apply the ring simulation in buzz model is visualized in Fig. 6. For making three-axis model, the adaptive step-ring selection is more suitable as it avoids tool retraction before nonlinear touch of geometry away from collision given the vertical machining tool orientation toward the model. The step-ring selection based on optimized model has a few pass loops by design so that we can have safe estimate of depth-based material cutting in each path iteration.

The series of ring selection is shown in Fig. 6, and the steps are 21, 24, 27, 33, 34, and 121 rings for each iteration due to its planned movement and 3D surface coverage. The ring distribution is not symmetric with the edge line touching boundary as the first touch of ring with less width margin to the edge or for steeper part of geometry across the ring horizontal plane that the model is located on.

The area of ring selection may cause retraction path when the edge points get too beyond ring cover area in path planning. To maximize ring number, we increase the ring number when ring coverage is not enough to cause retraction while we reduce the ring number when the movement triggers retraction or collision in linear path.

The ring size can be useful for pattern selection to highlight a surface region with user interest of partial model surface path planning with minimal material removal.

### Accessibility Map Generation for Path Planning in GPU Distributed Parallel Computing.

What we consider for the result is the feasibility of step ring-based path planning and comparison with global and local features while generating the accessible map-based orientation based on accessibility space. The accessibility space is formulated by accessibility map sequence given the rotational calculation capability of a particular machine, and we formulate the path around this model surface geometry to give rough estimation of path feasibility and time of actual accessible path along the surface model which we generate. To understand the difference of ring path planning in local distribution with ring path planning in global distribution on full geometry model, we compare accessible map features in Results of Simulation and Accessibility Map section.

## Storage Efficiency by Path Partition

### Adaptive Partitioning.

The decomposition number ND satisfies the adaptive partition condition and optimal numerical metrics for path planning in digital form for decision making of workload distribution. Comparison variable selections majorly focus on depth, step distance, and model resolution to have adaptive partition based on optimization characteristic. The end purpose is to attain code storage economics and data efficiency in smooth motion, so that we get compressed data or compact G-code density to fit storage limit and machining range requirement in safe zone.

### G-Code Optimization by Partition and Compression.

Due to the limit of G-code file size in most CNC machine, we have to do postprocessing for raw G-code output so that we can have G-code size within the file limit. To have compressed G-code, we combine the G-code points that happen to be on the same line due to detailed sampling into accumulated pass points so that saved movement of G-code can be attained with bigger step move rather than a few smaller steps along the same line while machining. The file sizing in finishing phase before the G-code compression is over limit about 4–5 MB compared to the actual machine limit while the current machine only holds 2 MB G-code size. The graph displays the G-code partition view after the optimization that divides the large G-code file into multiple subfiles.

Take G-code size as 5 MB, for example, the limit of buffer storage is 2 MB for maximum processing data, and then we can pick ND = 2, for example, the full roughing visualization of G-code simulation path is shown in Fig. 7, and the complete roughing shape is shown.

For finishing path generation from smaller step distance, as we increase the density of path to have smooth surface quality, the G-code oversizing for machine becomes a problem to consider. Thus, path partition helps solve this problem by allowing each partitioned file as a distributed workload to fall within the machine storage boundary. In order to have reduced G-code subfile that fits the size limit of machine requirement, we partitioned G-code file size into subdivision within accessible limit. The first division of multidirection path view is displayed in Fig. 8 while the second division is shown in Fig. 9.

By adaptive path partitioning, we can satisfy storage limit for individual G-code by adaptively changing the ND in Eq. (2). By merging the G-code step points in the same direction of motion path, the G-code compression of buzz model is reduced by more than 30%.

## Grid Classification Via Adaptive HDT Depth for 3D Model Topology Analysis in Accessible Region

The orientation of the model sometimes is not as what we originally generated from the image as we have to fit it in a machine desirable way based on the machine axis platform and pick whatever orientation mapping the software selection. To have 3D grid generation of model-based path for freeform selection, we first run 3D grid to have resolution depth-based analysis on the model geometry. The model element complexity of geometry will have different levels of topology depth in space, leading to different color region in adaptive grid for classification. The displayed model structure with HDT grid classification is shown in Fig. 10.

Improper selection of model level sets or path cutting zone of a model may cause duplicate path loops when there is concaved region hole that gets trapped in the same path plane or low machining sensitivity to run G-code output to be accessible by available machine motion. Thus, the optimal sizing of model is highly important besides software correct path simulation. In reality, we consider both simulation of path planning and actual machining on CNC platform for accuracy of path mapping to an accessible model region.

When the 3D tree depth is above 3, we can highlight the region of model and get body level set partition by displaying the model regions in different colors. Take buzz model, for example, when the 3D tree depth is higher than 3, the buzz starts to have clear classification of body partition by topology. The display of different connected parts of buzz model such as buzz body, buzz arm, and buzz antenna show the distributed various height planes of different buzz geometry components that are connected in classified regions for path planning plane partition. In Fig. 10, only body part that is connected on the same height plane is displayed with the same color; in this way, we can get classified color for different height levels to indicated region connectivity for model partition so that a clear surface geometry of height classification can be generated just from the 2D image on the horizontal plane. As we get information of the body height level set partition clearly from the image, we can then set the no cutting zone plane based on the information we collect from 2D image data. The setting of no cutting zone plane on *z* coordinate with height Z_no_cut should respect the model height level set so that the no cutting zone of the model partition divides the cutting part above the zone plane machined and below the zone plane part not machined. For example, one arm of buzz is on Z_low and another arm of buzz is on Z_high, and the planning plane condition that satisfies three-axis path planning is Z_hight <= Z_no_cut, when the path planning will cut into the model with full complete cover within the limit of half-hemisphere for the machine tool travel. Bad planning condition such as Z_low <= Z_no_cut <= Z_high leaves holes of the multilevel set plane of model as no cutting plane has gaps not machinable in three-axis path planning. When Z_hight <= Z_no_cut, the tool path does not have to travel under the arm region.

Figure 10 shows the depth kernel-based 3D view of buzz model while HDT kernel depth distance varies from 1 to 3. From the graph, we can see the geometry shape of bee model mainly has two parts of its body geometry classification partitioned into red and green, while the model boundary is marked with yellow contouring line surrounding the upper body. More tree grid depth depicts the red colored shape as outside of body geometry while one single set of arm, antenna, and wing is on the same surface within the major body, and the rest of antenna and the other wing are ignored in path planning due to three-axis setting of no tool rotation. This is how we reference and pick the optimal cutting depth with no cutting zone plane in three-axis path planning to respect model geometry and machine capability of three-axis setting.

For model analysis, we pick the buzz model and the head model in Fig. 11 for comparison view in 3D volume to have detailed look of how two models are relatively different in size and geometry from each other. The topology of the model gets accurate display in kernel grid via the relative size of grid distribution. The grid is even distributed so that the actual grid blocks reflect the shape and measurement of 3D voxel model.

Model differentiation is feasible through grid partitioning and measurement based on key body intersection with horizontal grid plane. The dimension of the model is indicated by grid kernel layout and the topology of object gets highlighted with color classification as the 3D tree depth gets larger.

## Overplanning in Software Compared to Accessible Space Limit by Physics

Overplanning or overdecomposition of cutting depth: break total cutting depth of path planning into more iteration pieces than actual machining need, or assign each cutting iteration with cutting depth larger than required that the accumulated cutting depth of all iterations is bigger than maximum cutting accessibility of physic limit. This happens between software path planning and physical machining in practice during an unmatched planning.

## Results of Simulation and Accessibility Map

### Three-Dimensional Material and Path Simulation.

The 3D scallop simulation results shown by proper selection of the simulations match actual machining output. The simulation of each path iteration is based on the cutting depth preselected. Figure 12 demonstrates the accumulated path iterations, simulated 3D visualization of model geometry and actual machining product.

### Accessibility Map Computation Via GPU.

The map generation time between local ring number estimation and global all path automatic planning is compared from Figs. 13–17 for accessibility map analysis in different aspects. Tables 1 and 2 show highlighted comparison of step ring-based path planning and its optimal time of GPU for generating map. The GPU and CPU time displayed in the table is the time of accessible sequence computation based on number of points via different processor of GPU or CPU, and the map sequence is computed on GPU after workload of building cylinder and cone intersection of model processing through HDT. The number of point difference in map is significant for small model except when the model geometry is complex and in large scale. For model with high geometry complexity or large scale, the step ring-based path planning has benefit of generating local optimal path for minimal material removal. The results also show relatively lower efficiency for the first iteration of all path planning.

The current system platform used for the paper result generation is Windows 7 Enterprise running with Visual Studio(VS)2010 with GTX 780ti GPU and GTP 980ti GPU from Nvidia, and the processor is Intel Core (TM) CPU i7-4771 3.5 GHZ 3.5 GHZ, with installed memory 32 GB on 64-bit operating system. High multicore speed processors may have less rendering time for path generation and map processing.

## Conclusions

In this paper, we describe the data structure and mapping relation of model-driven system work flow of path planning with model validation through simulation via double feedback logic for path parameter tuning in subtractive 3D printing. By analyzing a model topology in grid classification in 2D image slices, no cutting region of path planning plane gets determined for model topology analysis in multilayered level sets. The model-driven workload by 3D geometry structure in multilevel set of adaptive grid and HDT-based rendering through varying cut depth gives the hint of optimal partition of model based on horizontal plane and model height classification. Adaptive ring distribution selection and ring size prediction are also described to avoid linear path retraction for collision-free path planning. Moreover, the storage efficiency is demonstrated for the G-code path optimization, and the modeling of parallel GPU simulation of accessibility map sequence is analyzed in path adaptive and scalable form.

To conclude, our major contribution is the integration of ring path selection and CAD modeling based on accessibility map generation in data analytics via different processor of CPU or GPU for time comparison on various GPU simulations. Data analytic aspects include accessible map relation with adaptive depth path, HDT branch relation, and end volume generation. We demonstrate time saving effects of step ring-based path planning in local versus global scale of all path automated planning with software user-oriented interaction. We also studied the grid adaptive feature on HDT in model structure analysis by topology. The results provide insights into high-performance computing and GPU parallelism and scalability of optimal path decision. In future, we aim at cloud computing and grid modeling feature with predictive data analysis for optimal path planning in subtractive 3D printing and 3D manufacturing.

## Acknowledgment

This work was supported by the National Science Foundation (NSF) under CPS Synergy Award No. 1329742.