Barrier regularization method for nonlinear grid distortion problem

Abstract

The invention discloses a barrier regularization method for a nonlinear grid distortion problem. The analysis steps of grid distortion suppression based on unit geometric mapping specifically comprise: obtaining input data of an analysis model, and initializing nonlinear analysis parameters; obtaining a Jacobian determinant based on the deformation gradient, and constructing a barrier regularization function; constructing a correction potential energy functional containing a barrier regular term, assembling the material stiffness matrix, the geometric stiffness matrix and the barrier stiffness matrix to obtain a tangent stiffness matrix, and solving a nonlinear equation to update a displacement field; if the displacement field is not convergent, updating the load step length and iteratively returning to calculate the deformation gradient, and if the displacement field is convergent, outputting the displacement; and constructing a topological optimization mathematical model, calculating an objective function value and sensitivity according to a displacement field obtained by convergence, updating unit density, adjusting material distribution or topological form, and outputting a final optimization structure. According to the method, unit distortion is directly suppressed in the analysis process, the numerical stability is enhanced, and the topological optimization convergence performance is improved.

Description

Technical Field

[0001] The present invention relates to the technical field of computational mechanics and structural optimization, and in particular to a barrier regularization method for nonlinear grid distortion problems. Background Art

[0002] In recent years, numerical analysis, structural optimization, and multi-physics simulation technologies that consider nonlinearity have been widely used in complex engineering design scenarios such as aerospace, intelligent manufacturing, 4D printing, and biomedicine. When geometric nonlinearity is involved, mesh distortion severely restricts computational stability and accuracy. Traditional numerical methods such as finite element and isogeometric methods often suffer from stiffness matrix singularities, numerical integration error accumulation, and solver convergence difficulties in large deformation analysis due to element Jacobian degeneration, element flipping, and severe distortion, which in turn affect subsequent structural optimization design. Existing technologies often rely on small stiffness adjustments or energy interpolation to alleviate these problems. However, these technologies suffer from the fact that small stiffnesses lack physical meaning, weakening the credibility of the results; they only alleviate stiffness singularities without directly controlling geometric distortion; and their applicability is limited, making them difficult to generalize.

[0003] In the process of geometric nonlinear analysis, common problems include unit mesh distortion, Jacobian degradation, and the resulting numerical instability. Therefore, there is an urgent need for an optimization technology that can suppress unit flipping and mesh degradation and improve the numerical stability of nonlinear numerical analysis and structural optimization. Summary of the Invention

[0004] In order to overcome the problems of mesh distortion, cell flipping and numerical instability that are common in large deformation analysis and geometric nonlinear structure optimization in the existing technology, the present invention provides a barrier regularization method for nonlinear mesh distortion problems. The present invention introduces a barrier regularization term based on the Jacobian determinant into the system potential energy functional, thereby directly suppressing cell distortion during the analysis and optimization process, enhancing numerical stability and improving the convergence performance of topology optimization.

[0005] In order to achieve the above object, the present invention adopts the following technical solutions:

[0006] As a preferred technical solution, it includes an analysis step of suppressing mesh distortion based on unit geometry mapping and a topology optimization step;

[0007] The analysis steps for mesh distortion suppression based on element geometry mapping include:

[0008] Obtain input data of the analysis model and initialize nonlinear analysis parameters;

[0009] Calculate the deformation gradient and Jacobian of each unit under the current step size, and construct a barrier regularization function based on the unit Jacobian;

[0010] A modified potential energy functional containing a barrier regularization term is constructed, and second-order variation of the modified potential energy functional is performed to obtain material stiffness matrix, geometric stiffness matrix and barrier stiffness matrix, and the tangent stiffness matrix is assembled by combining the stiffness matrices, and the displacement field is updated by solving the nonlinear equation;

[0011] The convergence of the displacement field is judged, if not convergent, the load step is updated and the calculation of the deformation gradient is returned for iteration, if convergent, the displacement is output, and the topology optimization step is entered;

[0012] In the topology optimization step, a topology optimization mathematical model is constructed, the objective function value and sensitivity are calculated according to the converged displacement field, the unit density is updated, the material distribution or topology morphology is adjusted, and the final optimized structure is output.

[0013] As a preferred technical solution, the input data of the analysis model is obtained, including initial geometry, material parameters and boundary conditions.

[0014] As a preferred technical solution, the nonlinear analysis parameters are initialized, including load step, iteration tolerance and barrier regularization parameter.

[0015] As a preferred technical solution, the deformation gradient and its Jacobian determinant of each element at the current step are calculated, specifically including:

[0016] The deformation gradient is composed of the derivative of the current displacement field with respect to the initial configuration coordinates, and is obtained by the shape function derivative of each element and the node displacement;

[0017] The Jacobian determinant is expressed as:

[0018] J = det(F)

[0019] Wherein, F represents the deformation gradient, and det represents the determinant of the matrix.

[0020] As a preferred technical solution, the barrier regularization function is a logarithmic barrier form, which is specifically expressed as:

[0021] ψ(J) = -γlog(J)

[0022] Wherein, ψ(J) represents the logarithmic barrier term, γ represents the barrier regularization parameter, and J represents the Jacobian determinant.

[0023] As a preferred technical solution, a modified potential energy functional containing a barrier regularization term is constructed, which is specifically expressed as:

[0024]

[0025] Where ∏(u) represents the modified potential energy functional, Ω0 represents the volume domain of the object in the reference configuration, W represents the strain energy density function, F represents the deformation gradient, u represents the displacement, V represents the volume domain of the structure or material in the reference configuration, b represents the body load per unit reference volume, and Γ t represents the boundary to which the surface force is applied, t represents the surface force load per unit reference area, A represents the surface area on the boundary, and ψ(J) is the barrier regularization function.

[0026] As an optimal technical solution, the second-order variation of the modified potential energy functional is obtained to obtain the material stiffness matrix, geometric stiffness matrix and barrier stiffness matrix, which are specifically expressed as:

[0027] K mat =∫ Ω B T DXF

[0028] K geo =∫ Ω G T SGV

[0029] K bar =∫ Ω φ(J)B T BdV

[0030] B=F G G

[0031] Among them, K mat represents the material stiffness matrix, K geo represents the geometric stiffness matrix, K bar represents the barrier stiffness matrix, F G represents the deformation gradient related terms matrix, G represents the shape function derivative matrix, V represents the volume domain of the structure or material in the reference configuration, S represents the second Piola-Kirchhoff stress tensor, φ(J) represents the second-order variation of the barrier regularization function ψ, and Ω represents the volume domain.

[0032] As a preferred technical solution, the various stiffness matrices are assembled to obtain the tangent stiffness matrix, which is specifically expressed as:

[0033] K=K mat +K geo +K bar

[0034] Where K represents the tangent stiffness matrix, K mat represents the material stiffness matrix, K geo represents the geometric stiffness matrix, K bar represents the barrier stiffness matrix;

[0035] The Newton-Simpson iteration is used to solve the nonlinear equation to update the displacement field. The nonlinear equation is expressed as:

[0036] KU=R

[0037] Among them, U is the displacement vector and R is the residual vector.

[0038] As an optimal technical solution, a topology optimization mathematical model is constructed, which is specifically expressed as follows:

[0039]

[0040] st:R(ρ,u)=0

[0041]

[0042] Among them, O represents the optimized objective function value, L represents the external load vector, u represents the displacement, ρ is the design variable vector, R is the residual vector, g represents the volume constraint function, and n e is the total number of discrete elements in the design domain, e represents the discrete elements in the design domain, v e represents the volume of the unit, ρ e Represents the cell density, V g The target volume of the structure is initially set.

[0043] As a preferred technical solution, the objective function value and sensitivity are calculated based on the converged displacement field. The objective function includes the output displacement of the compliant mechanism, minimization of structural flexibility, or maximization of buckling load.

[0044] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0045] (1) The present invention adopts a Jacobian determinant penalty mechanism based on unit geometry mapping. By introducing a barrier regularization function based on the Jacobian determinant into the system potential energy functional, the distorted mesh area is actively penalized. This effectively suppresses unit flipping and mesh distortion without introducing artificially small stiffness parameters, significantly improving the numerical stability and computational efficiency in the process of nonlinear structural analysis and topology optimization.

[0046] (2) The present invention adopts a logarithmic barrier regularization term with physical significance, which can be seamlessly embedded in nonlinear geometric analysis and large deformation topology optimization processes without modifying the basic material interpolation or optimization algorithm. It ensures the geometric consistency under high-order continuous basis functions, significantly improves the mesh quality, and reduces the problems of stiffness matrix morbidity and decreased integral accuracy caused by Jacobian degeneration.

[0047] (3) The present invention unifies the optimization models under material nonlinearity and geometric nonlinearity by constructing an enhanced potential energy form with a barrier term. It is not only applicable to classical structural optimization problems such as compliant mechanisms and beams, but also has good scalability and is compatible with traditional finite element and isogeometric methods. It has strong adaptability and broad application prospects.

[0048] (4) The present invention realizes the “vacuum hole” modeling capability and suppresses the numerical instability of the distorted area through the barrier term, rather than relying on the means of applying a small stiffness value to the airspace in the traditional variable density method, thereby fundamentally improving the physical rationality of the topology optimization solution and the stability of the optimization path.

[0049] (5) The present invention solves the numerical divergence problem caused by severe distortion of the spatial grid in existing nonlinear topology optimization, especially maintaining the stable convergence of the optimization process under large loads and large deformations. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 Schematic diagram of the process of the barrier regularization method for nonlinear grid distortion problem of the present invention;

[0051] Figure 2 It is a structural schematic diagram of the C-shaped beam of the present invention;

[0052] FIG3( a ) is a diagram showing the displacement and mesh deformation of the beam without using the barrier regularization method in the present invention;

[0053] FIG3( b ) is a diagram showing the displacement and mesh deformation of the beam using the logarithmic barrier regularization method of the present invention;

[0054] FIG3( c ) is a diagram showing the deformation process of the beam using the logarithmic barrier regularization method of the present invention;

[0055] Figure 4 It is a structural schematic diagram of the cantilever beam of the present invention;

[0056] FIG5( a ) shows the topology optimization configuration of the present invention without using the barrier regularization method;

[0057] FIG5( b ) shows the topology optimization configuration of the present invention using the logarithmic barrier regularization method. DETAILED DESCRIPTION

[0058] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0059] Example 1

[0060] like Figure 1As shown, this embodiment provides a barrier regularization method for nonlinear mesh distortion problems. By introducing the Jacobian determinant barrier regularization term into the system potential energy functional, mesh distortion control and numerical stability enhancement are achieved during the analysis and optimization process. The method includes a mesh distortion suppression analysis step and a topology optimization step based on cell geometry mapping. The specific implementation is as follows:

[0061] Analysis Step A:

[0062] Step A1: Read the input basic data of the isogeometric, finite element and other analysis models, including initial geometry, material parameters and boundary conditions;

[0063] Step A2: Initialize nonlinear analysis parameters, including load step size, iteration tolerance, and barrier regularization parameter, and enter analysis iteration;

[0064] Step A3: Calculate the deformation gradient F and its Jacobian J = det(F) of each element at the current step size, where the deformation gradient F is composed of the derivative of the current displacement field with respect to the initial configuration coordinates and can be obtained through the shape function derivatives and node displacements of each element;

[0065] Step A4: construct a barrier regularization function ψ(J) based on the unit Jacobian;

[0066] In this embodiment, the barrier regularization function ψ(J) includes a logarithmic or other continuously differentiable form.

[0067] Step A5: Construct a modified potential energy functional including the barrier regularization term, and calculate the second-order variation of the modified potential energy functional to obtain the material stiffness matrix K mat , geometric stiffness matrix K geo and the barrier stiffness matrix K bar ;

[0068] In this embodiment, the barrier stiffness matrix K bar The effect is only significant on severely distorted elements, avoiding excessive punishment of normal elements, ensuring the stability and convergence of nonlinear analysis, and thus ensuring the physical rationality of topological evolution;

[0069] Step A6: Assemble the material stiffness matrix K mat , geometric stiffness matrix K geo and the barrier stiffness matrix K bar The tangent stiffness matrix of is obtained and the nonlinear equations are solved to update the displacement field;

[0070] Step A7: Use the Newton-Simpson iteration method to determine the convergence of the displacement field. If it does not converge, update the load step size and return to step A3. If it converges, proceed to optimization step B.

[0071] Optimization step B:

[0072] Step B1: Initialize topology optimization parameters, including design variables, volume constraints, and optimization iteration times, and construct a topology optimization mathematical model. Specifically, topology optimization methods include but are not limited to density method, level set method, and evolutionary method.

[0073] Step B2: Calculate the objective function value and sensitivity based on the converged displacement field;

[0074] Specifically, the objective function includes, but is not limited to, minimizing the output displacement of the compliant mechanism, minimizing the structural flexibility, or maximizing the buckling load;

[0075] Step B3: Update the design variables, i.e., unit density, and adjust the material distribution or topology;

[0076] Specifically, the design variable updating methods include but are not limited to the optimization criterion method and the moving asymptote method, and at the analysis level, the Jacobian determinants of the airspace and the intermediate region in the topological evolution are guaranteed to be positive definite;

[0077] Step B4: Determine the convergence of the optimization. If the preset convergence tolerance is met, the final configuration is output. Otherwise, return to analysis step A2 and continue the analysis-optimization coupling iteration. If convergence is achieved, the final optimized structure and analysis results are output.

[0078] In this embodiment, the regularization parameters of step A2 and step B1 can be adaptively and dynamically adjusted according to the degree of mesh distortion to balance analysis stability and optimization convergence.

[0079] In this embodiment, the analysis and optimization of step A6 and step B2 are alternately cycled to improve the computational efficiency and numerical stability of the nonlinear problem;

[0080] The method of this embodiment is applicable to material nonlinear analysis, including models such as hyperelastic, elastoplastic and viscoelastic materials, and the barrier regularization term does not change the material constitutive relationship; it is applicable to geometric nonlinear analysis, including large displacement, large deformation and large strain scenarios, and the barrier regularization term is adaptively adjusted to prevent the Jacobian determinant from degenerating to a negative value; it is applicable to the numerical calculation framework of arbitrary dimensions, arbitrary grid types and arbitrary order shape functions, including isogeometry, finite element, etc.

[0081] The method of the present invention is applicable to various numerical analysis methods such as geometry and finite element analysis, as well as various structural optimization frameworks. It has wide adaptability to numerical platforms and versatility in optimization objects. It can significantly improve the computational stability and optimization convergence of large deformation nonlinear problems without introducing artificially small stiffness parameters.

[0082] Example 2

[0083] This embodiment takes a classic two-dimensional C-shaped beam as an example, verifies the grid distortion suppression effect of the barrier regularization method in geometrically nonlinear large deformation analysis based on the isogeometric analysis framework. The model geometry is shown in Figure 2 The beam size is 9 (long) x 9 (wide) x 1 (thick), the left side is fixed, the upper and lower ends of the beam are subjected to concentrated loads -2 and 2, and the material adopts a compressible Neo-Hookean model, and its strain energy density function is defined as:

[0084]

[0085] Where κ and χ are Lamé constants, C = F T F is the right Cauchy-Green tensor;

[0086] The barrier regularization method is compared with the non-barrier regularization method, where the grid in the void region is subjected to small stiffness in the non-barrier regularization method, and the regularization method is not subjected to small stiffness, to verify the improvement effect of the barrier regularization on the analysis convergence and grid quality.

[0087] The implementation steps of this embodiment are as follows:

[0088] Step S1: define the C-shaped beam geometry model, material parameters and boundary conditions, and set the barrier regularization parameters that can be adaptively adjusted according to the grid distortion degree:

[0089] γ = γ0(1 + 10(J max -J min ))

[0090] In the formula, γ0is the preset initial value, and γ0= 10 -4 , J max = max(det(F(ξ,η))), J min = max(min(det(F(ξ,η))),10 -6 ).

[0091] Step S2: initialize the load increment to 10, and set the nonlinear analysis convergence tolerance e tol = 10 -3 ;

[0092] Step S3: calculate the deformation gradient F of each element and the Jacobian determinant J = det(F).

[0093] Step S4: construct the barrier regularization function ψ(J) according to the element Jacobian determinant.

[0094] The threshold value δ needs to be defined (usually 0 < δ << 1). When the Jacobian determinant J = det(F) is lower than δ, it is set to J = δ, that is:

[0095] J = max(J, δ)

[0096] The threshold value δ of the embodiment is 10 -6 .

[0097] Log-type barrier term:

[0098] ψ(J) = -γlog(J)

[0099] Step S5: Construct a modified potential energy functional containing barrier regularization terms:

[0100]

[0101] Where Ω0 represents the volume domain of the object under the reference configuration; b represents the body force load per unit reference volume; t represents the surface force load per unit reference area; Γ t represents the boundary (part of the boundary) where the surface force is applied; u represents the displacement; ψ(J) is the barrier regularization function; V represents the volume domain of the structure or material under the reference configuration; A represents the surface area on the boundary.

[0102] Second-order variation of the potential energy functional can obtain the material stiffness matrix K mat , the geometric stiffness matrix K geo and the barrier stiffness matrix K bar , which are specifically represented as:

[0103] K mat = ∫ Ω B T DBdV

[0104] K geo = ∫ Ω G T SGdV

[0105] K bar = ∫ Ω φ(J)B T BdV

[0106] In the formula, B is composed of the deformation gradient related term matrix F G and the shape function derivative matrix G, and for a two-dimensional problem, it is defined as:

[0107]

[0108] In the formula, (u, v) represents the displacement field, (X, Y) represents the initial coordinates, and N represents the NURBS basis function. For a quadratic basis function, n = 9 is the number of control points.

[0109] Step S6: Assemble the tangent stiffness matrix K = K mat + K geo + Kbar , use Newton-Simpson iteration to solve the nonlinear equation KU=R, where U is the displacement vector and R is the residual vector, which is defined as the external force f ext and internal force f int difference;

[0110] Step S7: Determine the convergence tolerance. If not, return to S3 iteration. If converged, output the displacement.

[0111] The results are as follows Figures 3(a)-3(b) The following diagrams show the mesh deformation for the non-barrier regularization method and the logarithmic barrier regularization method, respectively. Under large deformation conditions, the internal mesh and the solid mesh of the unregularized mesh are severely distorted, making effective calculation impossible. However, the regularized method still achieves convergence, indicating that the barrier regularization term effectively prevents excessive mesh deformation and allows for smooth convergence. Figure 3(c) shows the mesh deformation process under logarithmic barrier regularization, which is smooth and reasonable.

[0112] Example 3

[0113] This example takes the cantilever beam topology optimization as an example to illustrate the classic solid isotropic material penalty method in topology optimization. For example, where E is the effective Young's modulus of the unit, ρ e represents the unit density, i.e. the design variable, β is the penalty factor, and E0 is the Young's modulus of the substrate. The optimization effect of the method of the present invention in geometric nonlinear topology optimization under the finite element framework is verified. The model geometry is as follows Figure 4 As shown, the cantilever beam is discretized into 120×30 units, fixed on one side, and a downward concentrated force of 10 is applied to the middle of the right end. 7 Taking the minimum flexibility of the structure as the goal and the volume fraction as the constraint set to 0.5, a topology optimization mathematical model is established:

[0114]

[0115] st:R(ρ,u)=0

[0116]

[0117] Among them, O represents the optimized objective function value, L is the external load vector, ρ is the design variable vector, and n e is the total number of discrete elements in the design domain, g represents the volume constraint function, v e Represents the volume of the unit, V g The target volume of the structure is initially set;

[0118] The implementation steps of this embodiment are as follows:

[0119] Step S1: Define the design domain, constraints, loads and other boundary conditions of the cantilever beam structure, define parameters such as volume fraction and grid size, and read and input the above data;

[0120] Step S2: Set the topology optimization initial variables, set the optimization convergence criteria, define the barrier regularization parameters and the nonlinear convergence criteria;

[0121] Step S3: Analysis-Optimization Cycle:

[0122] (a) Execute steps A2-A7 to complete the barrier regularization nonlinear analysis;

[0123] (b) Calculate the objective function and sensitivity;

[0124] (c) updating the design variables using the optimization criterion method;

[0125] (d) Check convergence. If not, return to (a).

[0126] The final topological configuration is Figures 5(a)-5(b) The following are the optimization results for the barrier-free and logarithmic barrier regularization methods, respectively. It is clearly observed that the topological configuration for the barrier-free method does not converge, indicating that the numerical tolerance limit has been exceeded under this load, while the barrier regularization method converges and achieves a reasonable configuration. The proposed method obtains an effective and reasonable topological configuration, further demonstrating its superiority in geometrically nonlinear topology optimization and its versatility across different numerical platforms.

[0127] The above embodiments are preferred implementation modes of the present invention, but the implementation modes of the present invention are not limited to the above embodiments. Any other changes, modifications, substitutions, combinations, and simplifications that do not deviate from the spirit and principles of the present invention should be considered as equivalent replacement methods and are included in the scope of protection of the present invention.

Claims

1. A barrier regularization method for nonlinear mesh distortion problems, characterized by: Includes analysis steps for suppressing mesh distortion based on unit geometry mapping and topology optimization steps; The analysis steps for mesh distortion suppression based on element geometry mapping include: Obtain input data of the analysis model and initialize nonlinear analysis parameters; Calculate the deformation gradient and Jacobian of each unit under the current step size, and construct a barrier regularization function based on the unit Jacobian; Construct a modified potential energy functional including the barrier regularization term, calculate the second-order variation of the modified potential energy functional to obtain the material stiffness matrix, geometric stiffness matrix and barrier stiffness matrix, assemble the various stiffness matrices to obtain the tangent stiffness matrix, and solve the nonlinear equation to update the displacement field; Determine the convergence of the displacement field. If it does not converge, update the load step size and return to calculate the deformation gradient for iteration. If it converges, output the displacement and enter the topology optimization step. In the topology optimization step, a topology optimization mathematical model is constructed, the objective function value and sensitivity are calculated based on the converged displacement field, the unit density is updated, the material distribution or topological morphology is adjusted, and the final optimized structure is output.

2. The barrier regularization method for nonlinear mesh distortion problem according to claim 1, characterized in that: Obtain input data for the analysis model, including initial geometry, material parameters, and boundary conditions.

3. The barrier regularization method for nonlinear mesh distortion problem according to claim 1, characterized in that: Initialize the nonlinear analysis parameters, including the load step size, iteration tolerance, and barrier regularization parameter.

4. The barrier regularization method for nonlinear mesh distortion problem according to claim 1, characterized in that: Calculate the deformation gradient and Jacobian of each unit under the current step size, including: The deformation gradient is composed of the derivative of the current displacement field with respect to the initial configuration coordinates, and is obtained through the shape function derivatives of each element and the node displacement; The Jacobian determinant is expressed as: J=det(F) Where F represents the deformation gradient and det represents the determinant of the solution matrix.

5. The barrier regularization method for nonlinear mesh distortion problem according to claim 1, characterized in that: The barrier regularization function is a logarithmic barrier form, specifically expressed as: ψ(J)=-γlog(J) where ψ(J) represents the logarithmic barrier term, γ represents the barrier regularization parameter, and J represents the Jacobian determinant.

6. The barrier regularization method for nonlinear mesh distortion problem according to claim 1, characterized in that: Construct a modified potential energy functional including the barrier regularization term, which is specifically expressed as: Where ∏(u) represents the modified potential energy functional, Ω0 represents the volume domain of the object in the reference configuration, W represents the strain energy density function, F represents the deformation gradient, u represents the displacement, V represents the volume domain of the structure or material in the reference configuration, b represents the body load per unit reference volume, and Γ t represents the boundary to which the surface force is applied, t represents the surface force load per unit reference area, A represents the surface area on the boundary, and ψ(J) is the barrier regularization function.

7. The barrier regularization method for nonlinear mesh distortion problem according to claim 1, characterized in that: The material stiffness matrix, geometric stiffness matrix and barrier stiffness matrix are obtained by taking the second-order variation of the modified potential energy functional, which can be expressed as: K mat =∫ Ω B T DBDV K geo =∫ Ω G T SGdV K bar =∫ Ω φ(J)B T BdV B=F G B Among them, K mat represents the material stiffness matrix, K geo represents the geometric stiffness matrix, K bar represents the barrier stiffness matrix, F G represents the deformation gradient related terms matrix, G represents the shape function derivative matrix, V represents the volume domain of the structure or material in the reference configuration, S represents the second Piola-Kirchhoff stress tensor, φ(J) represents the second-order variation of the barrier regularization function ψ, and Ω represents the volume domain.

8. The barrier regularization method for nonlinear mesh distortion problem according to claim 1, characterized in that: The tangent stiffness matrix is ​​obtained by assembling the various stiffness matrices, which is specifically expressed as: K=K mat +K geo +K bar Where K represents the tangent stiffness matrix, K mat represents the material stiffness matrix, K geo represents the geometric stiffness matrix, K bar represents the barrier stiffness matrix; The Newton-Simpson iteration is used to solve the nonlinear equation to update the displacement field. The nonlinear equation is expressed as: KU=R Among them, U is the displacement vector and R is the residual vector.

9. The barrier regularization method for nonlinear mesh distortion problem according to claim 1, characterized in that: Construct a topology optimization mathematical model, which is specifically expressed as follows: Among them, O represents the optimized objective function value, L represents the external load vector, u represents the displacement, ρ is the design variable vector, R is the residual vector, g represents the volume constraint function, and n e is the total number of discrete elements in the design domain, e represents the discrete elements in the design domain, v e represents the volume of the unit, ρ e Represents the cell density, V g The target volume of the structure is initially set.

10. The barrier regularization method for nonlinear mesh distortion problem according to claim 1, characterized in that: The objective function value and sensitivity are calculated based on the converged displacement field. The objective function includes the output displacement of the compliant mechanism, minimization of structural flexibility, or maximization of buckling load.

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