WO2017080135A1

WO2017080135A1 - 3d printing-oriented model decomposition and arrangement method

Info

Publication number: WO2017080135A1
Application number: PCT/CN2016/079151
Authority: WO
Inventors: 陈宝权; 贝纳斯佰德瑞驰; 陈学霖; 科恩奥丹尼尔; 胡瑞珍; 黄其兴; 林金杰; 吕琳
Original assignee: 山东大学
Priority date: 2015-11-10
Filing date: 2016-04-13
Publication date: 2017-05-18
Also published as: CN105427374A; CN105427374B

Abstract

Proposed is a 3D printing-oriented model decomposition and arrangement method, comprising: giving a model S and a printing space PV; calculating an initial pyramid attribute block decomposition of the model S, and performing voxelization on these pyramid attribute blocks to generate a first initial intermediate solution isolu₀; initializing an intermediate solution set ISolu ＝ {isolu₀}; initializing an optimal complete solution P^*; with regard to each intermediate solution isoli_i in the intermediate solution set ISolu, calculating a series of candidate intermediate solutions generated by the intermediate solution, and solving an optimal solution of decomposition and arrangement of the model S; and performing local optimization on a three-dimensional mesh of a plurality of extracted blocks, and by means of the translation of each block in the negative direction of the Z axis, arranging these blocks to be more compact, thereby obtaining the optimal decomposition and arrangement regarding the model S. A model is decomposed into fewer blocks so as to be effectively arranged into a printing space, so that the printing process is effectively carried out.

Description

A model decomposition and alignment method for 3D printing

Technical field

The invention relates to an optimization method for decomposition and arrangement of a model for 3D printing, in particular to an optimization method for segmentation and arrangement of a model based on model approximate pyramid decomposition and directed search.

Background technique

3D printing, a kind of rapid prototyping technology, is a technique of constructing an object by layer-by-layer printing based on a digital model file using a bondable material such as powdered metal or plastic.

In recent years, more and more 3D printing has been mentioned, and the ability to print digital model files into physical objects has made it widely used in many industries. From a technical point of view, 3D printing usually superimposes printed materials "layer by layer" to turn the digital blueprint into a real object. And such a 3D printing process is limited to a printing platform with a certain space. Therefore, the size of the model and the printing platform is directly related to whether the finished product can be printed at one time. When the overall size of the model exceeds the size of the print space, it cannot be completely placed in the print platform for printing. This requires us to decompose the model and then arrange the various parts of the model into the print space for printing. In response to this problem, Hu (HU, R., LI, H., ZHANG, H., AND COHEN-OR, D. 2014. Approximate pyramidal shape decomposition. ACM Trans. on Graph 33, 6, 213: 1–213:12 In 2014, et al. proposed a method that can decompose the model into as few blocks as possible satisfying the approximate pyramid properties, and then separate these blocks that satisfy the approximate pyramid properties and print them using a fused deposition rapid prototyping (FDM) printer. It can save support materials very effectively. However, this method only considers the decomposition of the model into blocks of approximate pyramid shape, and does not consider how to arrange these blocks into the print space for efficient printing. Luo(LUO,L.,BARAN,I.,RUSINKIEWICZ,S.,AND MATUSIK,W.2012.Chopper:Partitioning models into 3D-printable parts.ACM Trans.on Graph 31,6,129:1–129:9) etc. The method proposed by Man in 2012 can decompose a 3D model into a number of blocks, each of which can be placed in the print space for separate printing. The factors considered in this method are the number of blocks, the assemblability and the aesthetics of the cutting line. However, this method does not take into account the constraints of the support material or printing time. Vanek (VANEK, J., GARCIA, J., BENES, B., MECH, R., CARR, N., STAVA, O., AND MILLER, G. 2014. PackMerger: A 3D print volume optimizer. Computer Graphics Forum 33, 6, 322–332) A method proposed by others in 2014, considering how to better decompose the model can make the alignment result better. The method first extracts the surface shell of the model, and does not process it for the solid model. Then, the extracted shell model is tetrahedralized and then adopted. The bottom-up search strategy first decomposes the model and then uses the blocks obtained by the decomposition to arrange. The problem with this method is that it first decomposes the model into fixed blocks and then uses them to arrange them, which leads to a large reduction in the search space, which greatly reduces the possibility that the algorithm can find the optimal solution. . The method is decomposed and then arranged. We hope that when solving the problem of decomposition and arrangement of the model, the decomposition and the arrangement can be performed simultaneously, so that the model can be traversed and the arrangement and arrangement of the model can be traversed. Greatly increased. Since this is an NP-hard problem, we hope to find an approximate optimal solution within a limited search time. Considering that the current 3D printing and forming process is mainly divided into two types from the need of supporting structure in the molding process, one is based on the powder material bonding spray method, represented by 3DP, and the other is based on resin and plastic materials. The molding process represented by FDM. In the molding process, the former does not need a supporting structure, the overall printing time depends on the height of the model in the forming direction, the printing material depends on the cost of the three-dimensional model itself; the latter requires a supporting structure, the overall printing time and materials are supported by the three-dimensional model itself and The materials are determined together. The method disclosed in the present invention is applicable to the above two types of molding processes, and different constraints can be considered separately. In addition, in the process of decomposition and alignment, we hope that the model is decomposed into fewer blocks to facilitate the reassembly of the model later.

Note: The pyramid shape definition, which satisfies the shape of the pyramid attribute, means that the shape has such a bottom edge (bottom surface), and the vertical line of the bottom edge (bottom surface) is formed from any point inside the shape. Any point is inside the shape, and the shape that satisfies the above properties is what we call the pyramid shape.

Summary of the invention

In order to solve the above problems, the present invention proposes an optimization method for model decomposition and alignment for 3D printing. Based on the model's approximate pyramid decomposition and directed search strategy, the method can decompose the model into fewer blocks and arrange them into the print space according to the constraints of different printing technologies, which is suitable for powder printers and fused deposition rapid prototyping printers.

A model decomposition and alignment method for 3D printing, comprising the following steps:

Step 1: Given a model S, print the space PV, calculate the initial pyramid attribute block decomposition of the model S, and voxelize these pyramid attribute blocks to generate the first initial intermediate solution isolu ₀ ;

Step 2: Initialize the intermediate solution set ISolu={isolu ₀ }, and initialize the optimal complete solution P ^* ;

Step 3: Calculate a series of candidate intermediate solutions generated by the intermediate solution for each intermediate solution isolu _i in the intermediate solution set ISolu, and add the candidate intermediate solution set CSolu;

Step 4: Calculate the local objective function value for each csolu _i belonging to the candidate intermediate solution set CSolu, and select the k candidate intermediate solutions csolu ₀ ,..., csolu _k-1 with the highest score, and make the candidate intermediate Solution set CSolu={csolu ₀ ,...,csolu _k-1 };

Step 5: For each csolu _i belonging to the intermediate solution of the candidate intermediate solution set CSolu candidate, if the candidate intermediate solution csolu _i is a complete solution, calculate the objective function value, if the objective function value of the candidate intermediate solution csolu _i is greater than P ^* The objective function value, then P ^* = csolu _i ; otherwise discard the candidate intermediate solution csolu _i ;

If the candidate intermediate solution csolu _{i is} not yet a complete solution and the objective function value of the intermediate solution is greater than or equal to the objective function value of P ^* , then the candidate intermediate solution csolu _{i is} added to the intermediate solution set ISolu;

Step 6: If the intermediate solution set ISolu is not empty, jump to step 3; if the intermediate solution set ISolu is empty, then P ^{* is} the optimal solution solved, according to the block inside the print space in P ^* Arrangement, extracting the voxelized block actually corresponds to the three-dimensional mesh of the model S, that is, solving the decomposition and permutation optimal solution of the model S;

Step 7: Locally optimize the three-dimensional meshes of the plurality of blocks extracted in step six, and perform the panning in the negative direction of the Z-axis of each block to make the blocks more compact, thereby obtaining the model S. Optimal decomposition and alignment.

The specific steps in the first step are:

(1-1) Decompose the model S into the pyramid attribute block, and transform the orientation of these pyramid attribute blocks so that the bottom surface is parallel to the xoy plane, and voxelization results in a series of blocks {p ₀ , p ₁ , p ₂ , .. .};

(1-2) Initialize the intermediate solution isolu _0. In isolu ₀ , p ₀ , p ₁ , p ₂ , ... are not arranged in the print space PV.

The specific steps in the second step are:

(2-1) Initialize the intermediate solution set, so that ISolu={isolu ₀ };

(2-2) Initialize the optimal complete solution P ^* such that the objective function value of P ^* is an infinite minimum.

The specific steps of calculating the candidate intermediate solution in the third step are:

(3-1) Initializing the candidate intermediate solution set CSolu is empty;

(3-2) For each isolu _i belonging to the intermediate solution set ISolu, in the intermediate solution, the blocks {p ₀ , p ₁ , p ₂ , ...} are not arranged into the print space PV, for each A block produces a series of candidate intermediate solutions when arranged at each specific location in the print space;

(3-3) Add all candidate intermediate solutions generated in the step (3-2) to the candidate intermediate solution set CSolu.

Further, in step (3-2), each block is arranged in a complete arrangement to a specific position of the printing space or in a partial arrangement to each specific position of the printing space;

For each block p _i , it can be rotated 0°, 90°, 180° or 270° around the x, y or z axis and then moved in the form of a moving window at the bottom of the print space, ie the block p _{i is} arranged to print Each specific position of the space produces a series of candidate intermediate solutions. In this case, a block outside the print space is completely arranged into the print space, which is called a complete arrangement;

In addition, a block p _i can also be divided into two blocks p _i0 , p _i1 , p _i0 or p _i1 by one cross or vertical cut by 0°, 90°, 180° or 270° around the x, y or z axis. And then move in the form of a moving window at the bottom of the print space, ie, block p _i0 or p _{i1 is} arranged at each specific position of the print space, resulting in a series of candidate intermediate solutions, in which case one is in the print space The outer block has only the divided pieces that are arranged into the print space, and the remaining parts form a single block, which is left outside the print space and is called a partial arrangement.

The method for calculating the local objective function value of a candidate intermediate solution in the fourth step is:

Assume that there is already a heap D consisting of some arranged blocks inside the current print space. One candidate intermediate solution is to arrange a block p into the print space, form a new heap D' with D, and define a block for evaluating the block. The local objective function of p arranged into heap D:

n: the current candidate intermediate solution all the number of blocks, including the arrangement or the number of all blocks arranged into the print space;

v:

The voxels in the new pile D'formed;

p, D forms the number of voxels in all vertical gaps in D', these vertical gaps will be temporarily filled by the powder material during the printing process;

H _v is the height of voxel v in the print space, H _PV is the height of the entire print space;

α, η: User input specified, adjustable parameters.

The calculation method of the objective function value for a complete solution in the step 5 is:

N: completely solves all the blocks in the print space;

H (PV): the height of the print space;

H(D): completely solves the height of the stack D in which all the blocks in the print space are arranged;

α: User input specified, adjustable parameters.

The above algorithm is suitable for powder printers.

Further, the above description algorithm can be applied to the fused deposition rapid prototyping printer by changing the objective function in the fifth step to the following objective function calculation method:

V(S): the number of all voxels after voxelization of all blocks in step one;

G _FDM (D): the number of voxels for all vertical gaps in heap D;

N: completely solves all the blocks in the print space;

α: User input specified, adjustable parameters.

The invention has the beneficial effects that the printing model is too large to be printed in the printing space, and a new algorithm is proposed. According to the printing constraints of different printers, the model is decomposed into fewer blocks and effectively arranged into the printing space. To make the printing process work effectively.

The method of the present invention is based on model approximate pyramid decomposition and directed search strategy, can decompose the model into fewer blocks, and is arranged into the printing space according to the constraints of different printing technologies, and is suitable for powder printers and fused deposition rapid prototyping printers. .

Note: The intermediate solution, the intermediate solution is an incomplete solution, consisting of the print space and the blocks outside the print space. In the intermediate solution, although there are some blocks that have been arranged inside the print space, there are at least one block. Not arranged into the print space.

Complete solution, unlike the intermediate solution, all blocks have been arranged inside the print space.

DRAWINGS

Figure 1 is a cross-sectional view of a tetrahedralized model;

The initial solution is shown in Figure 2, consisting of some blocks on the left that are not aligned into the PV and an empty PV;

Figure 3 is a diagram showing the intermediate solution after a complete arrangement from the initial solution described in Figure 2;

Figure 4 is a diagram showing the intermediate solution formed after the intermediate solution of Figure 3 has been partially aligned;

In Fig. 5, the optimal solution obtained by the result of the intermediate solution shown in Fig. 4 is further arranged in one step;

Shown in Figure 6 is a mesh corresponding to the model S extracted from the voxelized block;

As shown in FIG. 7, it is the final result obtained by the partial optimization of step (7) of FIG. 6;

Figure 8 is a specific flow chart of the algorithm.

detailed description:

The invention will be further described below in conjunction with the drawings and embodiments. For ease of understanding and explanation, the details of the algorithm are shown in part in two-dimensional drawings.

In order to achieve the decomposition and alignment of the models suitable for powder printers and fused deposition rapid prototyping printers, the present invention adopts the following technical solutions (the following description takes a powder printer as an example, and finally gives a method of how to apply the method to a fused deposition rapid prototyping printer. description of)

The specific flow chart of this algorithm is shown in Figure 8.

(1) Given a model S, print the space PV, calculate the initial pyramid attribute block decomposition of S, and voxelize these pyramid attribute blocks to generate the first initial intermediate solution isolu ₀ (in this intermediate solution, all The blocks are not arranged into the PV);

(2) Initialize the intermediate solution set ISolu={isolu ₀ }, initialize the optimal complete solution P ^* ;

(3) For each intermediate solution in the ISolu solution isolu _i , calculate a series of candidate intermediate solutions, and add the candidate intermediate solution set CSolu; empty

(4) For each csolu _i belonging to CSolu, calculate its score (local objective function value), and select the k candidate intermediate solutions csolu ₀ ,..., csolu _k-1 with the highest score, and make CSolu={csolu ₀ ,...,csolu _k-1 };

(5) For each csolu _i belongs to CSolu,

If csolu _i is a complete solution, calculate its objective function value, if csou _i 's objective function value is greater than the P ^* objective function value, then P ^* = csolu _i ; otherwise discard csou _i ;

If csolu _{i is} not yet a complete solution and the objective function value of the intermediate solution is greater than or equal to the objective function value of P ^* , add csolu _i to ISolu, ISolu=ISolu+{csolu _i };

(6) If the ISolu is not empty, then jump to step (3); if the ISolu is empty, then the P ^* at this time is the optimal solution solved, according to the arrangement of the blocks in the print space in P ^* , The extracted voxelized block actually corresponds to the three-dimensional mesh of S, that is, the decomposition and the optimal solution of S are solved.

As shown in Fig. 5, it is the optimal solution obtained by the result of the intermediate solution shown in Fig. 4 and then completely arranged in one step. The solution consists of three blocks, and the result of the arrangement is very low, which is advantageous for accelerating the printing process of the powder printer.

Shown in Figure 6 is a mesh corresponding to the model S extracted from the voxelized block.

(7) The three-dimensional mesh of the plurality of blocks extracted in step 6 is locally optimized, and the blocks are arranged more tightly by being translated in the negative direction of the Z-axis of each block. Thus, we get the optimal decomposition and arrangement of the model S.

As shown in Figure 7, this is the final result of Figure 6 after partial optimization of step (7).

The specific steps in the step (1) are as follows:

(1-1) Decompose the pyramid attribute block by S, and transform the orientation of these pyramid attribute blocks so that the bottom surface is parallel to the xoy plane, and voxelization results in a series of blocks {p ₀ , p ₁ , p ₂ ,... };

In Figure 1, a two-dimensional example is shown, with the initial input shape (and its The pyramid shape is decomposed), and the voxelization result of the pyramid shape that it decomposes (right side of the figure).

(1-2) Initialize the intermediate solution isolu ₀ , in isolu ₀ , p ₀ , p ₁ , p ₂ , ... are not arranged into the PV;

The initial solution is shown in Figure 2, consisting of some blocks on the left that are not aligned into the PV and an empty PV.

The specific steps in the step (2) are:

(2-1) Initialize the intermediate solution set, so that ISolu={isolu ₀ };

(2-2) Initializing the optimal complete solution P ^* , and let the objective function value of P ^* be an infinite minimum value;

The specific steps of calculating the candidate intermediate solution in the step (3) are:

(3-1) Initializing the candidate intermediate solution set to be empty CSolu;

(3-2) For each isolu _i belonging to ISolu, in the intermediate solution, blocks {p ₀ , p ₁ , p ₂ , ...} are not arranged into the PV.

For each block p _i , it can be rotated 0°, 90°, 180° or 270° around the x, y or z axis and then moved in the form of a moving window at the bottom of the print space (ie the block p _{i is} arranged to print) Each specific location of the space produces a series of candidate intermediate solutions. In this case, a block outside the print space is completely arranged into the print space, which we call a complete arrangement;

Figure 3 shows the intermediate solution after a complete alignment from the initial solution described in Figure 2.

In addition, a block p _i can also be divided into two blocks p _i0 , p _i1 by one cross cut or vertical cut (here we filter out the case where the split result is not two), p _i0 or p _i1 around x, y or The z-axis is rotated by 0°, 90°, 180° or 270° and then moved in the form of a moving window at the bottom of the print space (ie, the block p _i0 or p _{i1 is} arranged at each specific position of the print space), resulting in a series of Candidate intermediate solution. In this case, a block outside the print space has only the portion that has been sliced into the print space, and the remaining portion forms a separate block, which remains outside the print space, which we call a partial arrangement. .

Figure 4 shows the intermediate solution formed by the intermediate solution in Figure 3 after a partial arrangement.

(3-3) Add all candidate intermediate solutions generated in the step (3-2) to CSolu.

The calculation method of the local objective function value for a candidate intermediate solution in the step (4) is:

Assume that there is already a heap D consisting of some arranged blocks inside the current print space. One candidate intermediate solution is to arrange a block p into the print space to form a new heap D' with D. We define a local objective function for evaluating the arrangement of the block p into the heap D:

n: the current candidate intermediate solution all the number of blocks (including the arrangement or the number of all blocks arranged into the print space);

v:

The voxels in the new pile D'formed;

α, η: User input specified, adjustable parameters.

The calculation method of the objective function value for a complete solution in the step (5) is:

N: completely solves all the blocks in the print space;

H (PV): the height of the print space;

α: User input specified, adjustable parameters.

Finally, the above described algorithm can be applied to a fused deposition rapid prototyping printer by changing the objective function in step (5) to the following objective function calculation method:

V(S): the number of all voxels after voxelization of all blocks in step (1);

G _FDM (D): the number of voxels for all vertical gaps in heap D;

N: completely solves all the blocks in the print space;

α: User input specified, adjustable parameters.

The above description of the specific embodiments of the present invention has been described with reference to the accompanying drawings, but it is not intended to limit the scope of the present invention. Those skilled in the art should understand that the skilled in the art does not require the creative work on the basis of the technical solutions of the present invention. Various modifications or variations that can be made are still within the scope of the invention.

Claims

A model decomposition and alignment method for 3D printing, characterized in that it comprises the following steps:

Step 1: Given a model S, print the space PV, calculate the initial pyramid attribute block decomposition of the model S, and voxelize these pyramid attribute blocks to generate the first initial intermediate solution isolu 0 ;

Step 2: Initialize the intermediate solution set ISolu={isolu 0 }, and initialize the optimal complete solution P * ;

Step 3: Calculate a series of candidate intermediate solutions generated by the intermediate solution for each intermediate solution isolu i in the intermediate solution set ISolu, and add the candidate intermediate solution set CSolu;

Step 4: Calculate the local objective function value for each csolu i belonging to the candidate intermediate solution set CSolu, and select the k candidate intermediate solutions csolu 0 ,..., csolu k-1 with the highest score, and make the candidate intermediate Solution set CSolu={csolu 0 ,...,csolu k-1 };

Step 5: For each csolu i belonging to the intermediate solution of the candidate intermediate solution set CSolu candidate, if the candidate intermediate solution csolu i is a complete solution, calculate the objective function value, if the objective function value of the candidate intermediate solution csolu i is greater than P * The objective function value, then P * = csolu i ; otherwise discard the candidate intermediate solution csolu i ;

If the candidate intermediate solution csolu i is not yet a complete solution and the objective function value of the intermediate solution is greater than or equal to the objective function value of P * , then the candidate intermediate solution csolu i is added to the intermediate solution set ISolu;

Step 6: If the intermediate solution set ISolu is not empty, jump to step 3; if the intermediate solution set ISolu is empty, then P * is the optimal solution solved, according to the block inside the print space in P * Arrangement, extracting the voxelized block actually corresponds to the three-dimensional mesh of the model S, that is, solving the decomposition and permutation optimal solution of the model S;

Step 7: Locally optimize the three-dimensional meshes of the plurality of blocks extracted in step six, and perform the panning in the negative direction of the Z-axis of each block to make the blocks more compact, thereby obtaining the model S. Optimal decomposition and alignment.
A model for 3D printing-based model decomposition and arrangement according to claim 1, wherein the specific steps in the first step are:

(1-1) Decompose the model S into the pyramid attribute block, and transform the orientation of these pyramid attribute blocks so that the bottom surface is parallel to the xoy plane, and voxelization results in a series of blocks {p 0 , p 1 , p 2 , .. .};

(1-2) Initialize the intermediate solution isolu 0. In isolu 0 , p 0 , p 1 , p 2 , ... are not arranged in the print space PV.
A model for 3D printing-based model decomposition and arrangement according to claim 1, wherein the specific steps in the second step are:

(2-1) Initialize the intermediate solution set, so that ISolu={isolu 0 };

(2-2) Initialize the optimal complete solution P * such that the objective function value of P * is an infinite minimum.
A model for 3D printing-based model decomposition and arrangement according to claim 1, wherein the specific steps of calculating the candidate intermediate solution in the third step are:

(3-1) Initializing the candidate intermediate solution set CSolu is empty;

(3-2) For each isolu i belonging to the intermediate solution set ISolu, in the intermediate solution, the blocks {p 0 , p 1 , p 2 , ...} are not arranged into the print space PV, for each A block produces a series of candidate intermediate solutions when arranged in a particular location in the print space;

(3-3) Add all candidate intermediate solutions generated in the step (3-2) to the candidate intermediate solution set CSolu.
A 3D printing-oriented model decomposition and arrangement method according to claim 4, wherein, in the step (3-2), each block is arranged in a complete arrangement to each specific position of the printing space or Arranged in a partial arrangement to a specific location in the print space.
A 3D printing-oriented model decomposition and alignment method according to claim 5, wherein for each block p i , it can be rotated by 0°, 90°, 180° around the x, y or z axis or 270°, then moving in the form of a moving window at the bottom of the print space, ie, arranging the blocks p i to each specific position of the print space, producing a series of candidate intermediate solutions, in which case one is outside the print space The blocks are completely arranged into the print space, which is called a complete arrangement.
A model for 3D printing-based model decomposition and arrangement according to claim 5, wherein a block p i can also be divided into two blocks p i0 , p i1 , p i0 or p by one cross-cut or vertical cut. I1 is rotated by 0°, 90°, 180° or 270° around the x, y or z axis and then moved in the form of a moving window at the bottom of the print space, ie the block p i0 or p i1 is arranged to each specific print space Position, produces a series of candidate intermediate solutions, in which case a block outside the print space has only the segmented partial blocks arranged inside the print space, and the remaining portions form a single block, leaving Outside the print space, call it a partial arrangement.
A model for 3D printing-based model decomposition and arrangement according to claim 1, wherein the method for calculating the local objective function value of a candidate intermediate solution in the fourth step is:

Assume that there is already a heap D consisting of some arranged blocks inside the current print space. One candidate intermediate solution is to arrange a block p into the print space, form a new heap D' with D, and define a block for evaluating the block. The local objective function of p arranged into heap D:

n: the current candidate intermediate solution all the number of blocks, including the arrangement or the number of all blocks arranged into the print space;

v:
The voxels in the new pile D'formed;

p, D forms the number of voxels in all vertical gaps in D', these vertical gaps will be temporarily filled by the powder material during the printing process;

H GAIN (v, D):
H v is the height of voxel v in the print space, H PV is the height of the entire print space;

α, η: User input specified, adjustable parameters.
A model for 3D printing-based model decomposition and arrangement according to claim 1, wherein the calculation method of the objective function value for a complete solution in the step 5 is:

N: completely solves all the blocks in the print space;

H (PV): the height of the print space;

H(D): completely solves the height of the stack D in which all the blocks in the print space are arranged;

α: User input specified, adjustable parameters.
A model for 3D printing-based model decomposition and arrangement according to claim 9, wherein the description algorithm in the fifth step is changed to the following objective function calculation method, and the above description algorithm can be applied to the rapid deposition of fused deposition. Forming printer:

V(S): the number of all voxels after voxelization of all blocks in step one;

G FDM (D): the number of voxels for all vertical gaps in heap D;

N: completely solves all the blocks in the print space;

α: User input specified, adjustable parameters.