CN115344986B - Device and method for improving strength of three-dimensional structure - Google Patents
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Abstract
The application discloses a device and a method for improving the strength of a three-dimensional structure, comprising the following steps: reading a three-dimensional model and data of a stress configuration comprising a stress area; according to the read data, the three-dimensional model is subjected to high-precision discretization into a plurality of voxels, and a density variable is distributed to each voxel; calculating the worst load and displacement field for the three-dimensional model by using a multiple grid method and a power method; judging whether the strain energy of the three-dimensional model to the worst load is converged or not; and if the density variable is converged, generating result data of the three-dimensional model according to the density variable, otherwise, updating the density variable of each voxel according to the distribution and displacement field of the worst load, and returning to the step of calculating for recalculation. Compared with the traditional method, the method disclosed by the application is used for large-scale worst-case topology optimization, has a much higher speed, can greatly reduce time expenditure, and is suitable for modeling and design in manufacturing industry and building industry.
Description
Technical Field
The application relates to the field of three-dimensional modeling and structure optimization, in particular to a device and a method for improving the external load resistance of a three-dimensional structure by performing topology optimization under the large-scale worst condition so as to improve the strength of the three-dimensional structure.
Background
The topology optimization is a method for improving the external load resistance of the structure by optimizing the material distribution in the structure, and the topology-optimized structure can use fewer consumables to achieve similar or even better performance with the original solid structure, so that the cost is saved, and the method is widely applied to manufacturing industry and building industry. When the external load is uncertain, one topology optimization method is to optimize its strain energy at the worst load distribution. For numerical calculations, finite element dispersion is generally performed on the structure, and the higher the dispersion accuracy, the larger the problem size. After discretization, the worst load distribution can be converted into a matrix eigenvalue problem, and the calculation cost of the matrix is related to the number of stress points, and when the problem scale or the stress area is relatively large, the calculation cost of the matrix can face very large calculation cost. Another approach is to translate the problem into a semi-positive scheduling problem, and as the problem grows in size, the time and memory consumption of the semi-positive scheduling is also very high. Thus making this problem difficult.
There is no method that can significantly reduce the computational overhead of handling the above problems in large scale situations, and the computational time is unacceptable. For example, for a common 3D printed toy model, the stress on the surface of the model is uncertain, in order to improve the strength of the toy, prevent the toy from being damaged by playing, and save printing consumables, the worst-case topology optimization method can be used to optimize the structure inside the model, so as to strengthen the fragile place of the model to resist the uncertain load. Because the whole surface of the model is likely to be subjected to external force, the existing method is utilized to perform worst-case topological optimization under a relatively large stress area, and the calculation power of a general household computer can take weeks or even months, so that the working efficiency is unacceptable.
Disclosure of Invention
The application aims to provide a novel large-scale worst-case topology optimization device and method, which can obviously reduce the calculation cost, ensure that the calculation time of solving by the calculation power of a common household computer is still acceptable, and further improve the strength of a three-dimensional structure.
The application solves the corresponding problems by adopting the following technical scheme: the method comprises the following steps of performing high-precision worst-case topology optimization on an input three-dimensional structure through computer software, and reinforcing the three-dimensional structure according to an optimization result: reading a three-dimensional model and data of a stress configuration comprising a stress area; according to the read data, the three-dimensional model is subjected to high-precision discretization into a plurality of voxels, and a density variable is distributed to each voxel; calculating the worst load and displacement field for the three-dimensional model by using a multiple grid method and a power method; judging whether the strain energy of the three-dimensional model to the worst load is converged or not; and if the density variable is converged, generating result data of the three-dimensional model according to the density variable, otherwise, updating the density variable of each voxel according to the distribution and displacement field of the worst load, and returning to the step of calculating for recalculation.
According to the technical scheme provided by the application, the worst load distribution is efficiently solved by using a new power method, so that the defect of high time and calculation overhead in the traditional method is effectively avoided, and a user does not need to have related expertise. The calculation time of the optimization process at the same accuracy can sometimes be reduced from weeks to months to hours compared to the prior art. For example, the user only needs to provide a three-dimensional model and material parameters and stress configuration of a filling model, and the model can be subjected to high-precision worst-case topology optimization on a common home computer through the algorithm of the application, so that the model is convenient to use and is particularly suitable for manufacturing design and industrial buildings. The method is easy to integrate into CAD/CAE software, or is independently made into software for designers or engineers, and can finish optimizing the internal structure of the model with the designed appearance in a few hours by the calculation force of a household computer when the requirement exists, thereby improving the carrying capacity of the model on the external uncertain load. The patent can be applied to manufacturing industries such as toys and the like and building industries such as bridges and the like.
The enhanced strength of three-dimensional structures described in this patent includes, but is not limited to: the strength of the three-dimensional structure is enhanced by changing the material distribution of the three-dimensional structure; other three-dimensional structures of the same strength as the original three-dimensional structure, etc., are realized by less material in various cases as will occur to those of ordinary skill in the art.
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In order to more clearly illustrate the technical solutions of the embodiments of the present application, the drawings that are needed in the description of the embodiments will be briefly described below, it being obvious that the drawings in the following description are only some embodiments of the present application, and that other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a flowchart of an algorithm for large-scale worst-case topology optimization provided by an embodiment of the present application;
FIGS. 2 and 3 are schematic diagrams of input models and stress configurations and final optimization results according to embodiments of the present application; the left graph is the outline of the model, the solid arrow frames or the area pointed by the solid arrow is the stress area provided by the user, and the area pointed by the open triangle is the fixed boundary given by the user; the middle graph is an optimized density field graph; the right graph is a cross-sectional view of the solid model generated from the density field after the final optimization is completed.
Detailed Description
Preferred embodiments of the present disclosure will be described in more detail below with reference to the accompanying drawings. While the preferred embodiments of the present disclosure are shown in the drawings, it should be understood that the present disclosure may be embodied in various forms and not limited to the embodiments described herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the present disclosure to those skilled in the art.
The term "comprising" and variations thereof as used herein means open ended, i.e., "including but not limited to. The term "or" means "and/or" unless specifically stated otherwise. The term "based on" means "based at least in part on". The terms "one example embodiment" and "one embodiment" mean "at least one example embodiment. The term "another embodiment" means "at least one additional embodiment". The terms "first," "second," and the like, may refer to different or the same object for descriptive purposes only and are not to be construed as indicating or implying a relative importance or implicitly indicating the number of technical features indicated. Thus, a feature defining "a first" or "a second" may explicitly or implicitly include one or more such feature. Other explicit and implicit definitions are also possible below.
The order of the method steps recited herein does not necessarily indicate that the method must be performed in that order. The order of steps is not limiting unless it is explicitly believed that the skilled artisan (e.g., programmer) would directly recognize after reading this disclosure that the steps should be performed in a particular order. In modern computer systems, method steps may be performed in parallel, or in a different order than presented herein, as desired.
According to computer software protection regulations, software includes computer programs and their related documents. And a computer program refers to a sequence of coded instructions that can be executed by a computer to obtain a certain result, or a sequence of symbolic instructions or symbolic statements that can be automatically converted into a sequence of coded instructions; the document refers to text data, charts, etc. for describing the contents, composition, design, functional specifications, development condition, test result, and use method of the program.
Unlike the strict distinction in the above-mentioned regulations, the terms software and program may be used interchangeably herein, particularly in conjunction with the context. The code described herein is a string of characters representing information in discrete form, which may or may not form part of the software. The code may or may not run independently. The length and complexity of the code is typically less than that of the software or program.
The present embodiment disassembles part of the steps to describe the technical solution more clearly.
In the reading step, the three-dimensional model and the force configuration entered by the user, as well as the discrete accuracy selectable by the user, the material parameters E, V used and the desired material quantity V are read by a reading unit of the computer device * ∈[0,1]。
Here, E, v represents the modulus of elasticity and poisson's ratio of the filler material. For example, if a 3D printed resin model is optimized, these two parameters refer to the elastic modulus and poisson's ratio of the resin. V (V) * Representing the filling ratio required by the user, i.e. the volume of filling material inside the model to the external volume of the model. If a solid model is desired, this parameter is 1, if it is completely hollow, it is 0, and under other parameters, the algorithm of the present application optimizes the corresponding internal structure so that the filling ratio is exactly user-specified.
The force-receiving arrangement generally comprises: the force-receiving area, and the user-selectable fixed boundary and whether the external force is constrained to be surface normal.
Here, if the user does not input discrete accuracy, material parameters, material usage, fixed boundaries, constraints, or the like, algorithm default values may be used. These parameters are used in the subsequent processing. Additionally, while the force-receiving area is typically entered by a user, the force-receiving area may also be computer-generated. The force zone, while generally specified by the user, may not be generated within the algorithm, but if the user does not provide a force zone, the full surface of the model may be specified as the force zone default.
For example, a user may input a three-dimensional model of a structure, typically a mesh file, to a computer and may give the stress area and fixed boundaries of the model. As shown in the left-hand diagrams of fig. 2 and 3.
In the discrete step, the three-dimensional model is subjected to high-precision discrete by a discrete unit of a computer. High precision discretization as described herein generally refers to discretizing a three-dimensional model into 2000-3000 tens of thousands of voxel units, which for the high precision described in the art will be understood by those of ordinary skill to generally refer to precision on the order of tens of millions or more.
The computer discrete unit will discrete the three-dimensional model according to the accuracy given by the user, and each voxel in the grid will be assigned a density variable ρ e ∈[0.001,1]Initializing each density variable to v; and a series of coarsened grids are regenerated from the discretized grids according to the requirement of the multiple grid method.
Here ρ e Represents a density variable assigned to each voxel unit after finite element discretization of the model, which represents the degree of voxel filling material, 1 represents the voxel filling entity, and 0 represents the voxel as empty. To prevent numerical problems, the lower bound is typically specified to be 0.001 instead of 0.
In the calculation step, the optimization problem of the three-dimensional model is solved by a calculation unit of the computer.
On the discretized grid, the stress configuration describes the nodes and the stress direction in which the external force can occur. From this information, the constraint satisfied by the external force f can be written as nf=0. Good loadThe strain energy f which can be generated by the strain T K -1 Describing f, large strain energy means that the stiffness of the three-dimensional structure under this load is small, and vice versa. In order to compare different load distributions equally, it is necessary to define them to have the same size, so that the two modes of the external force are constrained to be 1, that is, iifii= 1, so that the worst load can be found by solving the following problems:
s.t.‖f‖＝1，
Nf＝0。
here, K represents the overall stiffness matrix of the model after voxelization. The integral stiffness matrix is assembled from the stiffness matrix of each voxel, which can be calculated asWherein p is typically 3 and K 0 Is a solid unit (ρ) e When=1). After voxelization, the vertex of each voxel is a node, each node is provided with an external force, f represents the discretized result of the stress on the surface of the three-dimensional model, and the external forces on the points form an integral external force vector f. N represents the three-dimensional model after discrete and post-conversion, only the stress area designated by the user is stressed, and the stress direction can be determined. The linear constraint nf=0 constituted by the matrix and f describes the conditions given by these users.
Let a set of orthonormal basis components of ker N constitute a matrix H, the above problem can be translated into one:
here, ker N, H refer to the null space of matrix N and its corresponding set of orthogonal basis, respectively.
This is equivalent to matrixAt this point, it is necessary to optimize the strain energy of the structure at the worst load for a given material usage, i.e., solve the problem:
s.t.∑ e ρ e ≤V * ∑ e 1，
wherein λmax Representing the maximum eigenvalue.
In the calculation step, in particular, the worst load distribution and the displacement field can be determined using a modified power method.
Can be obtained by a power method:
1. randomly generate a phi 0 And normalize phi 0 ：＝φ 0 /‖φ 0 ‖
2. Order theAnd normalize phi i+1 ＝φ i+1 /||φ i+1 ||
3. Judging convergence, if so, exiting and returning phi i+1 The method comprises the steps of carrying out a first treatment on the surface of the No let i: =i+1 and back to 2.
Here, phi, x, y represent the intermediate vectors used in the power iteration.
To avoid calculating H, the application introduces a new variable y i ＝Hφ i This way the iteration can be changed to y i+1 ＝HH T K - 1 y i Due to HH T Happens to be the projection to space kerN, denoted P. It can be expressed as p=i-N T N, thus each timeIn iteration, the application only needs to execute multiplication of the matrix N and the transposition thereof and the vector and one-time vector subtraction operation, thereby omitting H. While for efficient calculation of K -1 y i The application uses a multi-grid method and can perform parallel calculation with the display card. And only make one V-cycle at a time to obtain K -1 y i To enter the next power iteration. It has been found through practice that this can greatly reduce the time overhead and can also converge.
The power method comprises the following steps:
1. generating random initial force y on the finest grid 0 Projected to the feasible region and normalized. Initializing displacement field x 0 =0, and let i=0.
2. With current displacement field x i Calculating residual r i ＝||y i -Kx i I and x i As V-cycle of the primary multiple-grid method on the grid sequence, an updated displacement field is obtained:
x i+1 ＝x i +V -1 (r i )，
3. the updated displacement field x i+1 The force field projected to the feasible region space and normalized to obtain update is:
4. judgment r i <∈ r and ||yi+1 -y i ||<∈ f Whether or not they are simultaneously established, if so, exiting and returning to f worst ＝y i+1 As the worst load distribution sum u worst ＝x i+1 As the displacement field at the worst load distribution. If not, let i: =i+1 and returns to 2.
Here, V -1 Representing the V-cycle process of the primary multiple-grid method. f (f) worst ,u worst The external force and the displacement vector respectively correspond to the worst case, namely, the worst case is the case of generating the maximum strain energy.
In the judging step, the meterThe judging unit of the computer judges the strain energy under the calculated worst loadIf the convergence is not, the method exits, otherwise, the updating step is entered. After exiting, the result data of the three-dimensional model can be generated by a generating unit of the computer according to the optimized density variable, as shown in the middle diagrams of fig. 2 and 3. The three-dimensional model may be printed out from the result data by a 3D printer, as shown in the right diagrams of fig. 2 and 3, or may be produced by pouring cement from the result data in a factory, or may be produced in other known or future developed manners.
The power method has the variables x and y corresponding to displacement and load respectively, when the algorithm converges, y converges to the worst load f worst X converges to the corresponding worst displacement u under the load worst The worst displacement field is the deformation displacement of the model under the worst load. Their relationship is f worst ＝Ku worst
In the updating step, the updating unit of the computer updates the density variable based on the determined displacement field under the worst load.
And updating the density variable according to an optimal criterion method.
The sensitivity for each voxel is first calculated:
here, g e Representing the sensitivity defined on each voxel, the meaning of this value is the density variable ρ in that voxel e The magnitude of the influence of the variation of (g) on the worst-case strain energy e The greater the description of the worst strain energy vs ρ e The more sensitive the variation of (c).
The sensitivity is then filtered:
wherein ,
p i is the center coordinate of the ith voxel.
The update criteria for density are as follows:
wherein Clamp (x, L, U) is a clamping function, when x>Returning U when U, when x<And returning to L when L, otherwise returning to x.Is a multiplier obtained by dichotomy search, and the purpose of the search is to update the volume ratio V after the density is updated according to the above formula new ＝∑ e ρ e /∑ e 1 is close to the volume dose v given by the user: v new -v|<10 -4 . Δρ is the maximum step size of the density variable per update, and is typically set to 0.02 to 0.08.
And returning to the calculation step after updating the density variable.
Disclosed herein may be methods, apparatus, systems, storage media, and program products. The program product may be stored in a readable storage medium having computer instructions for performing various aspects of the present disclosure. These instructions, when executed by the processing unit of a computer or other programmable data processing apparatus, result in an apparatus that implements the functions/acts specified in the flowchart and/or block diagram block or blocks. These computer readable program instructions may also be stored in a computer readable storage medium that can direct a computer, programmable data processing apparatus, and/or other devices to function in a particular manner, such that the computer readable medium having the instructions stored therein includes an article of manufacture including instructions which implement the function/act specified in the flowchart and/or block diagram block or blocks.
The computer readable program instructions described herein may be downloaded from a computer readable storage medium to a respective computing processing device or to an external computer or external storage device over a network, such as the internet, a local area network, a wide area network, and/or a wireless network. The network may include copper transmission cables, fiber optic transmissions, wireless transmissions, routers, firewalls, switches, gateway computers and/or edge servers. The network adapter card or network interface in each computing processing device receives computer readable program instructions from the network and forwards the computer readable program instructions for storage in a computer readable storage medium in the respective computing processing device. Wireless transmission may use solutions for wireless communication such as Wireless Local Area Network (WLAN), bluetooth (BT), global Navigation Satellite System (GNSS), frequency Modulation (FM), near field wireless communication technology (NFC), infrared technology (IR), etc. of wireless fidelity (Wi-Fi) networks.
Computer program instructions for performing the operations of the present disclosure can be assembly instructions, instruction Set Architecture (ISA) instructions, machine-related instructions, microcode, firmware instructions, state setting data, or source or object code written in any combination of one or more programming languages, including an object oriented programming language such as Smalltalk, c++ or the like and conventional procedural programming languages, such as the C language or similar programming languages. The computer readable program instructions may be executed entirely on the user's computer, partly on the user's computer, as a stand-alone software package, partly on the user's computer and partly on a remote computer or entirely on the remote computer or server. In the case of a remote computer, the remote computer may be connected to the user's computer through any kind of network, including a Local Area Network (LAN) or a Wide Area Network (WAN), or may be connected to an external computer (for example, through the Internet using an Internet service provider). In some embodiments, aspects of the present disclosure are implemented by personalizing electronic circuitry, such as programmable logic circuitry, field Programmable Gate Arrays (FPGAs), or Programmable Logic Arrays (PLAs), with state information of computer readable program instructions, which can execute the computer readable program instructions.
Various aspects of the present disclosure are described herein with reference to flowchart illustrations and/or block diagrams of methods, apparatus, systems, storage media and program products according to embodiments of the disclosure. It will be understood that each block of the flowchart illustrations and/or block diagrams, and combinations of blocks in the flowchart illustrations and/or block diagrams, can be implemented by computer-readable program instructions.
In the embodiments described herein, it should be understood that the disclosed apparatus and methods may be implemented in other ways. For example, the apparatus embodiments described above are merely illustrative, e.g., the division of modules or units is merely a logical functional division, and there may be additional divisions when actually implemented, e.g., multiple units or components may be combined or integrated into another apparatus, or some features may be omitted or not performed. On the other hand, the coupling or direct coupling or communication connection shown or discussed with each other may be an indirect coupling or communication connection via some interfaces, devices or units, which may be in electrical, mechanical or other forms.
The units described as separate parts may or may not be physically separate, and the parts shown as units may be one physical unit or a plurality of physical units, may be located in one place, or may be distributed in a plurality of different places. Some or all of the units may be selected according to actual needs to achieve the purpose of the solution of this embodiment.
In addition, each functional unit in the embodiments of the present application may be integrated in one processing unit, or each unit may exist alone physically, or two or more units may be integrated in one unit. The integrated units may be implemented in hardware or in software functional units.
The integrated units, if implemented in the form of software functional units and sold or used as stand-alone products, may be stored in a readable storage medium. Based on such understanding, the technical solution of the embodiments of the present application may be essentially or a part contributing to the prior art or all or part of the technical solution may be embodied in the form of a software product stored in a storage medium, including several instructions for causing a device (may be a single-chip microcomputer, a chip or the like) or a processor to perform all or part of the steps of the methods of the embodiments of the present application.
The foregoing description of the embodiments of the present disclosure has been presented for purposes of illustration and description, and is not intended to be exhaustive or limited to the embodiments disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the various embodiments described. The terminology used herein was chosen in order to best explain the principles of the embodiments, the practical application, or the technical improvements in the marketplace, or to enable others of ordinary skill in the art to understand the embodiments disclosed herein. Various modifications and alterations of this disclosure will become apparent to those skilled in the art. Any modifications, equivalent substitutions, improvements, etc. that fall within the spirit and principles of the present disclosure are intended to be included within the scope of the present disclosure.
Claims (4)
1. A method for improving the structural strength of a three-dimensional model, comprising the steps of:
reading a three-dimensional model and data of a stress configuration comprising a stress area;
according to the read data, the three-dimensional model is subjected to high-precision discretization into a plurality of voxels, and a density variable is distributed to each voxel;
calculating the worst load and displacement field for the three-dimensional model by using a multiple grid method and a power method;
judging whether the strain energy of the three-dimensional model to the worst load is converged or not;
if the density variable is converged, generating result data of the three-dimensional model according to the density variable, otherwise, updating the density variable of each voxel according to the distribution and displacement field of the worst load, returning to the step of calculating for recalculation,
wherein, calculate the worst load and displacement field specifically is: taking the strain energy of the load under the worst condition as an objective function of the optimization problem, and constraining the external force two-mode to be 1 to obtain the optimization problem:
s.t.||f||＝1，
Nf＝0，
wherein f is an external force on a node, N represents a three-dimensional model after discretization, nf=0 is a constraint condition obtained according to stress configuration, K is an overall stiffness matrix of the three-dimensional model after voxelization,
converting the optimization problem into a maximum eigenvalue problem:
s.t.∑ e ρ e ≤V * ∑ e 1，
where H is the set of orthonormal bases of the null space of the matrix N, lambda max Representing the maximum eigenvalue, provided that the volume is limited, ρ e The density variable of each voxel unit after the three-dimensional model is subjected to finite element discretization, V * Representing the filling ratio required by the user, i.e. the volume of filling material inside the model to the external volume of the model,
the power method introduces the variable y i ＝Hφ i And will be opposite phi i Is changed to the iteration of y i Is a function of the iteration of (a):
will beSeen as a projection P to the null space of the matrix N,and gets explicit form->I.e. the iteration becomes:
y i+1 ＝PK -1 y i ，
approximating solution of K using multiple grid methods -1 y i Rather than doing an exact solution, i.e., doing the V-cycle once:
x i+1 ＝x i +V -1 (r i )，
wherein ri Is the current residual r i ＝y i -Kx i ，x i+1 Is to K -1 y i Is approximated by x i Representing the current displacement field, V -1 Representing the V-cycle process of the primary multiple-grid method,
the updating of the density variable is specifically: the sensitivity for each voxel is calculated:
wherein uworst Is the corresponding displacement field under worst load, and filters the sensitivity:
wherein ,
p i is the center coordinate of the ith voxel, and updates the density variable with the filtered sensitivity:
wherein Clamp (x, L, U) is a clamping function, returns U when x > U, returns L when x < L, otherwise returns x,is a multiplier obtained by dichotomy search, and the purpose of the search is to update the volume ratio V after the density is updated according to the above formula new ＝∑ e ρ e /∑ e 1 is close to the volume dose V given by the user: v (V) new -V|＜10 -4 Δρ is the maximum step size of each update of the density variable.
2. An apparatus for improving structural strength of a three-dimensional model, comprising:
a reading unit that reads the three-dimensional model and data including a force configuration of the force receiving area;
according to the read data, the three-dimensional model is subjected to high-precision discretization into a plurality of voxels, and a discrete unit of a density variable is distributed to each voxel;
a calculation unit for calculating a worst load and a displacement field for the three-dimensional model by using a multiple grid method and a power method;
a judging unit that judges whether or not the three-dimensional model converges on the strain energy of the worst load;
a generation unit for generating result data of the three-dimensional model according to the density variable when the judgment result of the judgment unit is convergence;
an updating unit that updates the density variable of each voxel according to the distribution of the worst load and the displacement field when the judgment result of the judging unit is non-convergence, and the calculating unit calculates again based on the density variable updated by the updating unit,
when the computing unit computes the worst load and the displacement field, the computing unit takes the strain energy of the load under the worst condition as an objective function of the optimization problem, and constrains the external force to be 1 in a two-mode to obtain the optimization problem:
s.t.||f||＝1，
Nf＝0，
wherein f is an external force on a node, N represents a three-dimensional model after discrete and post-formation, nf=0 is a constraint condition obtained according to stress configuration, K is an overall rigidity matrix of the three-dimensional model after voxelization,
and converts this optimization problem to a maximum eigenvalue problem:
s.t.∑ e ρ e ≤V * ∑ e 1，
where H is the set of orthonormal bases of the null space of the matrix N, lambda max Representing the maximum eigenvalue, provided that the volume is limited, ρ e The density variable of each voxel unit after the three-dimensional model is subjected to finite element discretization, V * Representing the filling ratio required by the user, i.e. the volume of filling material inside the model to the external volume of the model,
the power method used by the calculation unit introduces a variable y i ＝Hφ i And will be opposite phi i Is changed to the iteration of y i Is a function of the iteration of (a):
and will beViewed as a projection P to the zero space of the matrix N and given the explicit form +.>I.e. the iteration becomes:
y i+1 ＝PK -1 y i ，
the computing unit approximately solves for K using a multiple grid approach -1 y i Rather than doing an exact solution, i.e., doing the V-cycle once:
x i+1 ＝x i +V -1 (r i )，
wherein ri Is the current residual r i ＝y i -Kx i ，x i+1 Is to K -1 y i Is approximated by x i Representing the current displacement field, V -1 Representing the V-cycle process of the primary multiple-grid method,
the updating unit calculates a sensitivity corresponding to each voxel:
wherein uworst Is the corresponding displacement field under worst load, and filters the sensitivity:
wherein ,
p i is the center coordinate of the ith voxel, and updates the density variable with the filtered sensitivity:
wherein Clamp (x, L, U) is a clamping function, returns U when x > U, returns L when x < L, otherwise returns x,is a multiplier obtained by dichotomy search, and the purpose of the search is to update the volume ratio V after the density is updated according to the above formula new ＝∑ e ρ e /∑ e 1 is close to the volume dose V given by the user: v (V) new -V|＜10 -4 Δρ is the maximum step size of each update of the density variable.
3. A computer system comprising a processor and a memory, the memory being interconnected with the processor, the memory having instructions stored therein, the instructions when executed by the processor, performing the method of claim 1.
4. A computer readable storage medium, characterized in that the computer readable storage medium stores a computer program which, when executed by a processor, implements the method of claim 1.
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