A method for calculating crack propagation of a hollow lightweight foundation or dike partition plate

Abstract

This invention discloses a method for calculating crack propagation in hollow lightweight foundation pit or dam partition wall panels, relating to the interdisciplinary fields of geotechnical engineering and numerical computation. By obtaining the material, geometric, and boundary parameters of the partition wall panel, the irregular physical domain is mapped to a standard coordinate system using the proportional boundary finite element method to achieve geometric normalization. The isoparametric polygon shape function is derived using the separation characteristics of radial and circumferential variables. A hybrid integration scheme combining circumferential second-order Gaussian quadrature and radial semi-analytical solution is used to calculate the nonlocal integral of the bond basis near-field dynamics. The momentum equation is discretized, and the relationship between material points and elements is established. After iteratively solving for displacements, the nonlocal fracture criterion is used to capture the microcrack initiation, macrocrack propagation, and penetration process. This invention is adaptable to multi-cavity structures and semi-infinite domain foundation scenarios, balancing computational accuracy and efficiency, and providing technical support for structural safety assessment of foundation pit support and dam seepage prevention projects.

Description

Technical Field

[0001] This invention relates to the field of geotechnical engineering and numerical calculation, specifically to a method for calculating crack propagation in hollow lightweight foundation pits or dam partition walls. This method is applicable to the analysis of crack initiation, propagation, and penetration processes in hollow lightweight concrete slabs, honeycomb sandwich panels, and lightweight aggregate-filled slab foundation pits or dam partition wall components. Background Technology

[0002] In geotechnical engineering scenarios such as foundation pit engineering and dam engineering, hollow lightweight partition walls (such as hollow lightweight concrete panels, honeycomb / multi-cavity sandwich panels, lightweight aggregate or foamed material filled panels) are often used as retaining, separating, or seepage-proof components under conditions of weak foundations, proximity to water or rivers, or dewatering. Due to the thin-walled, multi-cavity, and densely connected structural characteristics of these components, stress concentration easily occurs at weak points in the panel (such as grouting joints and the edges of pores) under engineering loads such as excavation unloading, backfill loading, water level fluctuations, and wave impact, leading to the initiation of initial cracks that gradually propagate. Once the cracks penetrate, they not only significantly reduce the stiffness and load-bearing capacity of the component but may also increase seepage channels, causing safety risks such as seepage deformation and backfiltration failure. Therefore, accurate calculation of the crack propagation process of such components is a key technical requirement for ensuring the stability of foundation pit support and the reliability of dam seepage prevention.

[0003] Current numerical methods for crack propagation in hollow lightweight partition walls have significant limitations: the traditional finite element method (FEM), while adaptable to complex geometry and boundary conditions, requires frequent mesh re-division in crack propagation path capture and multi-crack interaction simulation, resulting in low computational efficiency. This is especially true when dealing with the coupling problem between multi-cavity partition wall structures and semi-infinite domain foundations, where excessively large mesh sizes can increase computational costs. The boundary element method, while reducing degrees of freedom and accurately handling infinite domains, has poor adaptability to material heterogeneity and nonlinear fracture behavior, making it difficult to reproduce the entire process of crack initiation and macroscopic penetration within hollow panels. The traditional near-field dynamics (PD) method, while naturally describing the fracture process, suffers from large nonlocal integration computations and complex boundary treatments, making it difficult to directly apply to large-scale numerical simulations of engineering-grade hollow partition walls. Furthermore, it lacks specific computational strategies for lightweight multi-cavity structures, failing to fully meet the actual engineering needs of foundation pits and dams.

[0004] In engineering practice, the scenarios of foundation pits and dams present a triple requirement for calculating crack propagation in hollow lightweight partition walls: high accuracy, high efficiency, and scenario adaptability. It is necessary to accurately capture the stress concentration effect and crack propagation law of multi-cavity structures while also considering computational efficiency under complex conditions such as semi-infinite domain foundations and heterogeneous materials. Existing methods often fail to simultaneously meet these requirements, leading to difficulties for engineers assessing the service safety of hollow lightweight partition walls. These difficulties include large discrepancies between calculated results and actual values, or excessively long calculation cycles that fail to support design decisions.

[0005] Therefore, there is an urgent need for an efficient and accurate crack propagation calculation technology that can closely fit the engineering scenarios of foundation pits and dams and is specially customized for hollow lightweight partition boards, so as to fill the gap between existing numerical methods and actual engineering needs. Summary of the Invention

[0006] Based on the above-mentioned technical problems, this application discloses a method for calculating crack propagation in hollow lightweight foundation pits or dam partition walls, specifically including:

[0007] Obtain the material parameters, geometric parameters, and boundary conditions of the hollow lightweight partition wall panel. The material parameters include elastic modulus, mass density, and Poisson's ratio. The boundary conditions include load boundaries and constraint boundaries.

[0008] Based on the scaled boundary finite element method, the irregular polygonal physical domain in the Cartesian coordinate system is mapped to the standard scaled boundary coordinate system, thereby realizing the geometric normalization of the physical domain;

[0009] By utilizing the separation property of radial and circumferential variables in the proportional boundary coordinate system, an isoparametric polygon shape function adapted to polygon geometry is derived to describe the potential field distribution in the physical domain.

[0010] A hybrid integration scheme combining circumferential second-order Gaussian quadrature and radial semi-analytical solution is adopted to calculate the nonlocal integral of bond basis peri-field dynamics;

[0011] Based on the aforementioned isoparametric polygon shape function and hybrid integral scheme, the near-field dynamic momentum equation of the discrete bond basis is used to establish the relationship between material points and element geometry.

[0012] Substitute the initial global stiffness matrix and iteratively solve for the displacement of material points. Combine this with the nonlocal fracture criterion to capture the initiation of microcracks, the propagation of macrocracks, and the penetration process within the partition wall.

[0013] Preferably, the physical domain geometric normalization is achieved through isoparametric mapping technology, and the coordinate interpolation formula for any point on the three-node line element is: in, For the circumferential coordinates of the proportional boundary coordinate system, The shape function of the three-node line element; any point on the three-node line element The coordinates are represented as: , ,in, , , , These are the Cartesian coordinates of the three nodes of the three-node line element;

[0014] The formula for the coordinates of any point within a triangular sector is: , ,in, The radial coordinates are those of the proportional boundary coordinate system. , Let be the coordinates of any point within the triangular sector in the proportional boundary coordinate system.

[0015] Preferably, the geometric transformation relationship between the Cartesian coordinate system and the proportional boundary coordinate system is derived based on the Jacobian matrix, which is: ,in, , They are respectively , right The partial derivatives, , They are respectively , right The partial derivatives, It is the Jacobian matrix of a three-node line element; , , Shape function right The partial derivatives, The determinant of the Jacobian matrix is: The mapping formula for the differential operator ∇ is: ,in, , , , This is the vector of coefficients for the differential operator mapping.

[0016] Preferably, the isoparametric polygon shape function is derived based on the Laplace equation, which is: ,in, Let be the potential function at a point within a triangular sector, used to describe the potential field distribution within the physical domain; the expression for the potential function is: , It is only related to the radial coordinate Related radial analytic functions, The shape function of the three-node line element;

[0017] The Euler-Cauchy differential equation is obtained by transformation using the weighted residual method: 0, where, , and These are the coefficient matrices, , They are respectively right The first and second partial derivatives, for The transpose of the matrix;

[0018] , , , It is a geometric interpolation matrix, and is related to... Irrelevant , Let be the vector of coefficients mapped by the differential operator. For shape function pairs The partial derivatives of .

[0019] Preferably, the second-order differential equation is transformed into a first-order differential equation: ,in, It is an analytic function related to internal flux. is the Hamiltonian matrix, used to reduce the order of second-order differential equations;

[0020] The expression for the Hamiltonian matrix Z is: ,in, , They are respectively , The inverse matrix; We obtain this through eigenvalue decomposition: ,in, It contains eigenvalues Transformation matrix of relevant eigenvectors, This represents the corresponding three-node line unit. Let be the integration constant, and , The eigenvector matrix, It is the potential at the node;

[0021] The expression for the shape function of an isoparametric polygon is: ,in, It is an isoparametric polygon shape function adapted to polygon geometry, used to describe the continuous distribution of potential field in physical domain.

[0022] Preferably, the momentum equation for the near-field dynamics of the bond group is: ,in, The mass density of the hollow lightweight partition wall panel. For material points At any moment acceleration vector, For material points The volume of the influence domain To influence any material point within the domain, , Material points , At any moment The displacement vector, For paired force functions, It is a volume force vector;

[0023] The equations for the paired force vectors are: ,in, For material points and position vector difference, For material points and The difference in displacement vectors, for The model;

[0024] The force function per unit volume of the prototype microelastic material is: ,in, For bond elongation, for The model, for The model, Here are the fracture parameters, and , This represents the critical bond elongation. For the micromodulus function, The radius of the influence domain.

[0025] Preferably, the micromodulus function The expression is: ,in, For the initial micromodulus, The influence domain weighting function; ,in, The elastic modulus of the hollow lightweight partition wall panel. Poisson's ratio; in, for The model;

[0026] time Material Point The damage variables are: ,in, As a damage variable, it is used to quantitatively characterize material points. At any moment The damage state is represented by the numerator as the volume integral of the broken bonds within the influence domain, and the denominator as the total volume of the influence domain.

[0027] Preferably, the momentum equation is discretized using the Galerkin weighted residual method, and the weak integral control equation is: ,in, To solve for the domain, For the weight function, It is a volume force vector. , Material points The volume element;

[0028] The approximate solution for displacement and the weighting function are as follows: , ,in, For the shape function matrix, This is a column vector of nodal displacements;

[0029] The weak-form equation after discretization of the Gaussian integral is: ,in, , These represent the number of Gaussian integration points within the solution domain and the influence domain, respectively. , These are the solution domain and the influence domain, respectively. The coordinates of a Gaussian point , These are the volume weights corresponding to the Gaussian points. for The transpose of .

[0030] Preferably, the matrix form of paired force vectors in the global coordinate system is as follows: ,in, As paired force vectors, , Material points exist directional force component, , Material points exist directional force component, For fracture parameters, For micro stiffness matrix, It is a displacement vector. , Material points exist Displacement in direction, , Material points exist Displacement in direction;

[0031] Microstiffness matrix in local coordinate system for: ,in, For the micromodulus function, for The model;

[0032] The transformation relationship between the global and local coordinate systems is as follows: ,in, Let be the paired force vector and displacement vector in the local coordinate system, respectively, and T be the transformation matrix. , , For material points Cartesian coordinates, , For material points Cartesian coordinates.

[0033] Preferably, the global microstiffness matrix is: ,in, Here is the microstiffness matrix of the local coordinate system. Transformation matrix The transpose of the matrix, , They are respectively , The square of, for and The product;

[0034] The bond stiffness matrix is: ,in, For material points For material points The bond stiffness matrix, For material points Volume weight, , Material points , The shape function;

[0035] The final matrix form of the momentum equation is: ,in, The global stiffness matrix. This is the global node displacement column vector. This is a global external load column vector used for iteratively solving for the displacement of material points.

[0036] Compared with the prior art, the technical solution of this application has the following technical effects:

[0037] This invention addresses the coupling characteristics of hollow slabs with thin walls, multiple cavities, and semi-infinite domain foundations. By standardizing the irregular physical domain using the proportional boundary finite element method and combining it with the nonlocal characteristics of near-field dynamics of bonded foundations, it accurately reproduces the stress concentration and crack initiation process in weak parts of the slab under conditions such as excavation unloading and water level fluctuations. This avoids the calculation deviations caused by insufficient scenario adaptation in traditional methods and provides a suitable numerical analysis tool for actual engineering scenarios.

[0038] This invention achieves a breakthrough in balancing computational accuracy and efficiency by employing a hybrid integration scheme that combines circumferential second-order Gaussian quadrature with radial semi-analytical solution. This scheme retains the semi-analytical characteristics of the proportional boundary finite element method, avoids radial discretization errors, and balances accuracy and cost through circumferential numerical integration. It can capture crack propagation paths without frequent mesh remapping, overcomes the low computational efficiency of traditional near-field dynamics calculations, and meets the need for rapid comparison of multiple schemes in engineering design.

[0039] This invention can capture the entire process of crack evolution, accurately tracking everything from microcrack initiation to macrocrack penetration. Based on isoparametric polygon shape functions and nonlocal fracture criteria, combined with iterative solutions of material point displacement, it can monitor the changes in the bonding state of material points across the entire domain in real time. It can clearly present the propagation direction, bifurcation morphology, and penetration mechanism of cracks in hollow slabs under load, providing comprehensive numerical basis for engineers to assess the degree of damage to partition walls and predict the risk of structural failure.

[0040] This invention directly serves the safety assessment and design optimization of projects such as foundation pit support and dam seepage prevention. Through precise crack propagation calculations, it can guide the reinforcement design of weak parts of the partition wall panel, reducing potential hazards such as seepage deformation and structural instability caused by crack penetration. The efficient calculation characteristics shorten the engineering design cycle and reduce the cost of numerical analysis, providing reliable technical support for the safe application of hollow lightweight partition walls in geotechnical engineering, and promoting the improvement of design and assessment levels in related engineering fields.

[0041] The above description is only an overview of the technical solution of this application. In order to better understand the technical means of this application and implement it in accordance with the contents of the specification, and to make the above and other objects, features and advantages of this application more obvious and understandable, the preferred embodiments of this application are described in detail below with reference to the accompanying drawings.

[0042] The above and other objects, advantages and features of this application will become more apparent to those skilled in the art from the following detailed description of specific embodiments in conjunction with the accompanying drawings. Attached Figure Description

[0043] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort. In all drawings, similar elements or parts are generally identified by similar reference numerals. In the drawings, the elements or parts are not necessarily drawn to scale.

[0044] Based on the description of the figures and their corresponding technical content in the document, the titles of the figures are as follows:

[0045] Figure 1 This is a flowchart illustrating the efficient numerical solution method for steady-state head field of a dam with complex foundation according to the present invention.

[0046] Figure 2 This is a schematic diagram of the boundary conditions of the hollow lightweight partition wall panel according to an embodiment of the present invention;

[0047] Figure 3 This is a schematic diagram illustrating the crack propagation of a triangular mesh hollow lightweight partition wall panel according to an embodiment of the present invention.

[0048] Figure 4 This is a schematic diagram illustrating the crack propagation of a quadrangular mesh hollow lightweight partition wall panel according to an embodiment of the present invention.

[0049] Figure 5 This is a schematic diagram of crack propagation in a polygonal mesh hollow lightweight partition wall panel according to an embodiment of the present invention. Detailed Implementation

[0050] To make the objectives, technical solutions, and advantages of the embodiments of this application clearer, the technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, not all embodiments. In the following description, specific details such as specific configurations and components are provided merely to help fully understand the embodiments of this application. Therefore, those skilled in the art should understand that various changes and modifications can be made to the embodiments described herein without departing from the scope and spirit of this application. In addition, for clarity and brevity, descriptions of known functions and structures are omitted in the embodiments.

[0051] It should be understood that the phrase "an embodiment" or "this embodiment" throughout the specification means that a specific feature, structure, or characteristic related to the embodiment is included in at least one embodiment of this application. Therefore, "an embodiment" or "this embodiment" appearing throughout the specification does not necessarily refer to the same embodiment. Furthermore, these specific features, structures, or characteristics can be combined in any suitable manner in one or more embodiments.

[0052] Furthermore, reference numerals and / or letters may be repeated in different examples within this application. Such repetition is for the purpose of simplification and clarity and does not in itself indicate a relationship between the various embodiments and / or settings discussed.

[0053] In this article, the term "and / or" is merely a description of the relationship between related objects, indicating that three relationships can exist. For example, A and / or B can mean: A exists alone, B exists alone, and A and B exist simultaneously. The term " / and" in this article describes another type of relationship between related objects, indicating that two relationships can exist. For example, A / and B can mean: A exists alone, and A and B exist alone. In addition, the character " / " in this article generally indicates that the related objects before and after it are in an "or" relationship.

[0054] In this article, the term "at least one" is merely a description of the relationship between related objects, indicating that there can be three relationships. For example, "at least one of A and B" can mean: A exists alone, A and B exist simultaneously, or B exists alone.

[0055] It should also be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion.

[0056] Example 1

[0057] This embodiment mainly describes a method for calculating crack propagation in hollow lightweight foundation pits or dam partition walls, such as... Figure 1 As shown, it specifically includes:

[0058] Obtain the material parameters, geometric parameters, and boundary conditions of the hollow lightweight partition wall panel. The material parameters include elastic modulus, mass density, and Poisson's ratio. The boundary conditions include load boundaries and constraint boundaries.

[0059] Based on the scaled boundary finite element method, the irregular polygonal physical domain in the Cartesian coordinate system is mapped to the standard scaled boundary coordinate system, thereby realizing the geometric normalization of the physical domain;

[0060] In the Cartesian coordinate system, for the potential field description problem of the physical field of a continuous medium, the Laplace equation is used as the core governing equation. First, the physical domain (i.e., the continuous medium region covered by the irregular polygonal element) and boundary constraints (such as potential function boundary values ​​and normal derivative boundary conditions) corresponding to the equation are defined. Based on the core idea of ​​the Scaled Boundary Finite Element Method (SBFEM) of "radial scaling and circumferential discrete interpolation," isoparametric mapping technology is introduced. By constructing the correspondence between the vertices of the polygonal element and the nodes of the standard scaled boundary coordinate system, the irregular polygonal physical domain is accurately mapped to the regular standard scaled boundary coordinate system, achieving the normalization of the physical domain's geometry and laying the coordinate foundation for the subsequent separation and solution of field variables.

[0061] By utilizing the separation property of radial and circumferential variables in the proportional boundary coordinate system, an isoparametric polygon shape function adapted to polygon geometry is derived to describe the potential field distribution in the physical domain.

[0062] After coordinate transformation, an interpolation basis function adapted to the polygon's geometric features (such as a high-order polynomial basis function that satisfies the continuity of polygon boundary interpolation) is selected. Taking full advantage of the separation of radial and circumferential variables in the scaled boundary coordinate system, the potential field variables are decomposed into a product of radial and circumferential functions. By satisfying the polygon element boundary interpolation conditions (the potential field value at the boundary nodes must match the interpolation results of the basis functions) and geometric compatibility requirements (the derivative of the shape function must match the gradient of the element's geometric deformation), the isoparametric polygon shape function is systematically derived. This shape function not only accurately reproduces the geometric contour of irregular polygon elements but also efficiently describes the continuous distribution characteristics of the potential field within the physical domain through interpolation operations, providing a core interpolation tool for subsequent coupled discrete solutions of near-field dynamics and scaled boundary finite element methods.

[0063] A hybrid integration scheme combining circumferential second-order Gaussian quadrature and radial semi-analytical solution is adopted to calculate the nonlocal integral of bond basis peri-field dynamics;

[0064] To address the issues of insufficient accuracy and low efficiency in nonlocal integral calculations of bond-basis peridynamics using polygonal elements, a proprietary hybrid integration scheme is introduced based on this core concept: Nonlocal integration essentially involves the cumulative calculation of all bond interactions within the influence domain of a material point. The circumferential integral corresponds to the summation of bond interactions along the polygonal element boundary direction. By discretizing the polygonal boundary into one-dimensional line elements and employing a second-order Gaussian quadrature formula, numerical integration is achieved, ensuring accuracy in bond interaction calculations while controlling computational costs. The radial integral corresponds to the accumulation of bond interactions within the radial influence domain of a material point in a scaled boundary coordinate system. This fully utilizes the semi-analytical characteristics of the scaled boundary finite element method (SBFEM) to analytically solve for radial variables, completely avoiding truncation and discretization errors introduced by radial numerical integration. Through the coupling of circumferential numerical integration and radial semi-analytical solution, accurate calculation of nonlocal integrals of bond interactions within polygonal elements is achieved, laying the foundation for the application of bond-basis peridynamics theory in polygonal elements and ensuring the stability and accuracy of subsequent calculations.

[0065] Based on the aforementioned isoparametric polygon shape function and hybrid integral scheme, the near-field dynamic momentum equation of the discrete bond basis is used to establish the relationship between material points and element geometry.

[0066] To solve the relevant equations, the continuous form of the bond-basis peri-field dynamics momentum equation needs to be discretized to the material point level for numerical solutions. The discretization process is based on the fundamental assumptions of bond-type peri-field dynamics and is implemented step-by-step, incorporating previous derivations: First, based on the isoparametric polygon shape function derived by the proportional boundary finite element method, the spatial distribution characteristics of material points within the polygonal element are adapted, and the relationship between the material point position and the element geometry is established through shape function interpolation; second, a hybrid integral scheme is used to accurately calculate the shape function values ​​and their derivatives, ensuring that the shape function satisfies the geometric compatibility of bonding between material points (bond length matching the shape function interpolation accuracy) and the nonlocal interpolation requirement (covering all bonded objects within the influence domain of the material points).

[0067] Substitute the initial global stiffness matrix and iteratively solve for the displacement of material points. Combine this with the nonlocal fracture criterion to capture the initiation of microcracks, the propagation of macrocracks, and the penetration process within the partition wall.

[0068] Based on the initial and boundary conditions, the initial global stiffness matrix of the final Peri-PolySBFEM is substituted into the solution for iterative material point displacement. Combined with the material point displacement field obtained by the iteration, the changes in the bonding state between material points in the whole domain are monitored in real time. Combined with the nonlocal fracture criterion of Peri-PolySBFEM, the initiation of microcracks, the propagation and penetration of macrocracks in hollow lightweight partition wall panels are gradually captured, realizing the integration of matrix solution of bond basis near-field dynamic momentum equation and structural damage evolution analysis.

[0069] Furthermore, isoparametric mapping technology is introduced to map the physical domain of irregular polygonal elements to a standard scale boundary coordinate system. This formulation not only preserves the basic properties of the finite element normal function but also exhibits superior adaptability to arbitrary polygonal meshes. The coordinates of any point on a three-node line element are obtained through interpolation:

[0070]

[0071] The coordinates of any point (x, y) on a three-node line element can be represented as:

[0072]

[0073]

[0074] in, , ;

[0075] By connecting the three-node line elements to the scaling center, a triangular sector can be obtained. Through numerical interpolation in the circumferential direction and analytical calculation in the radial direction, the coordinates of any point within this triangular sector can be obtained, as shown in the following equation:

[0076]

[0077]

[0078] Based on the chain rule of differentials, the geometric transformation relationship between the Cartesian coordinate system and the scaled boundary coordinate system can be derived:

[0079]

[0080] in, The Jacobian matrix for a triangular sector can be defined as follows:

[0081]

[0082] in, , , ;

[0083] It is a Jacobian matrix of three-node line elements, and its determinant is:

[0084]

[0085] It can be deduced that:

[0086]

[0087] Therefore, differential operators Can be mapped to a scaled boundary coordinate system:

[0088]

[0089] in, , .

[0090]

[0091] Here, It is the potential at the node.

[0092] Will By substituting the integration constant c into the potential function, the potential path at a point within the triangular sector can be derived.

[0093]

[0094] Here, the superscript (e) represents the corresponding three-node line element. Therefore, the shape function equations derived using the Scaled Boundary Finite Element Method (SBFEM) are:

[0095]

[0096] Furthermore, the shape function of the isoparametric polygon is obtained by transforming the Laplace equation into an Euler-Cauchy type equation in a scaled boundary coordinate system using the weighted residual method, and then obtaining the analytical radial and circumferential interpolation solutions through eigenvalue decomposition. This ensures that the shape function satisfies both boundary interpolation and geometric compatibility. The shape function of the scaled boundary finite element is obtained by deriving the Laplace equation, which is:

[0097]

[0098] in, Let be the potential (or potential function) at a point within a triangular sector, and its expression can be written as:

[0099]

[0100] in, It is a radial analytic function.

[0101] By transforming the shape functions of the scaled boundary finite element using the weighted residual method, the Euler-Cauchy differential equation can be obtained:

[0102]

[0103] Wherein, the coefficient matrix , and Specifically, it can be expressed as follows:

[0104]

[0105]

[0106]

[0107] Among them, here and and Irrelevant.

[0108]

[0109]

[0110] Transform the second-order differential equation in the Euler-Cauchy differential equation into a first-order differential equation:

[0111]

[0112] in, It is an analytic function related to internal flux. Z is the Hamiltonian matrix, and its expression can be written as:

[0113]

[0114] We obtain this through eigenvalue decomposition:

[0115]

[0116] Where V is a subset of... The transformation matrix of the relevant eigenvectors. c is the integration constant, whose value can be obtained using the formula:

[0117]

[0118] Here, It is the potential at the node.

[0119] Will Substituting the integration constant c into the potential function, the potential path at a point within the triangular sector can be derived:

[0120]

[0121] Here, the superscript (e) represents the corresponding three-node line element. Therefore, the shape function derived through the Scaled Boundary Finite Element Method (SBFEM) can be expressed as:

[0122]

[0123] Furthermore, the concept of bonds in near-field dynamics is used to assess damage to the sheet metal, as bond breakage directly characterizes material damage. At time t, the momentum equation for point x in the material is:

[0124]

[0125] Where ρ is the mass density.

[0126] Each bond generates a pair of force vectors between two connected material points, which are equal in magnitude and opposite in direction. The equations for the paired force vectors in BB-PD theory are as follows:

[0127]

[0128] For a prototype microelastic (PMB) material, the force function per unit volume of material point x acting on material point x′ within its domain is:

[0129]

[0130] This involves introducing a bond elongation equation and defining fracture parameters. :

[0131]

[0132]

[0133] In addition, the micromodulus function The expression is as follows:

[0134]

[0135]

[0136]

[0137] The damage state of material point x at time t can be quantitatively characterized by continuous damage variables, and its mathematical expression is:

[0138]

[0139] Furthermore, discretizing the near-field dynamic momentum equations of the bond basis and quasi-static problems will... and Substituting into the momentum equation, we get the following equation:

[0140]

[0141] The above equations are treated using the weighted residual (Galerkin) method, which involves multiplying both sides of the equation by the weight function v(x) to obtain the corresponding weak integral governing equations in the solution domain Ω:

[0142]

[0143] Since an exact solution to the displacement function u(x) is difficult to obtain, an approximate solution u that meets the accuracy requirements is used. h The approximate solution is obtained by interpolating the nodal displacements d using the shape function N(x). Simultaneously, the weight function v(x) is made to have the same form as the shape function. Therefore, the displacements and weight functions to be solved are as follows:

[0144]

[0145]

[0146] In subsequent calculations, the solution domain Ω matrix is ​​discretized into several elements. The element volume is... Substituting the displacements and weight functions to be solved into the weak integral control equations for the integration domain, we obtain the following equations:

[0147]

[0148] After discretizing the above equation using Gaussian integration, the weak form of the bond-basis peri-field dynamic momentum equation can be written as:

[0149]

[0150] in, It is the integral sum of all particles within the influence domain corresponding to particle x. This integral sum reflects the interaction between all Gaussian points within the influence domain, indicating that damage cracking can be simulated by breaking the bonds between Gaussian points.

[0151] Furthermore, an initial global stiffness matrix coupled with near-field dynamics and scaled boundary finite element method is formed for iterative solution of material point displacements. For paired force vectors in the global coordinate system, it can be represented in the following matrix form:

[0152]

[0153] Where k is the microstiffness matrix. The array forms of the paired force vectors u of f are as follows:

[0154]

[0155]

[0156] In the local coordinate system, the above three formulas can be expressed as:

[0157]

[0158]

[0159]

[0160] in Represented as:

[0161]

[0162] The relationship between f and u in the global and local coordinate systems is as follows:

[0163]

[0164]

[0165] The expression for the transformation matrix T is:

[0166]

[0167] The expressions for l and m are:

[0168]

[0169]

[0170] From the above formula, the micro-stiffness matrix can be obtained as follows:

[0171]

[0172] in, for volume, For point mass For point mass The applied micro-stiffness matrix; and These represent the nodal displacements within their respective elements, combined with the bond stiffness matrix. for:

[0173]

[0174] Therefore, the first term on the left-hand side of the near-field dynamic momentum equation of the bond group can be rewritten as:

[0175]

[0176] By performing a double-loop summation on the above equation, the element stiffness matrix can be obtained. After assembling according to the global node numbers, the global stiffness matrix is ​​formed. The final matrix form of the bond-basis near-field dynamic momentum equation is: where is the global stiffness matrix, and is the external load:

[0177]

[0178] In this embodiment, a structural model of a hollow lightweight partition wall panel under load is established. Based on the theory of elasticity and the basic equations of near-field dynamics, the nonlocal dynamic control equations of the partition wall panel are constructed. A polygonal element subdivision strategy is used to discretize the structural domain into polygonal meshes adaptable to complex geometries. Within each polygonal element, the radial semi-analytical properties and angular higher-order interpolation functions of the Scaled Boundary Finite Element Method (SBFEM) are introduced to approximate the displacement field through interpolation. Through coordinate transformation and the principle of virtual work, the structural domain control equations based on the Peri-PolySBFEM framework are derived. Subsequently, matrix processing and eigenvalue decomposition methods are used to solve the structural dynamic equations, further yielding the dynamic equilibrium equations considering nonlocal actions and boundary constraints.

[0179] In the solution process, the Galerkin discretization method and semi-analytical radial integration strategy are used to enable crack propagation simulation and dynamic response analysis to be performed within a unified framework. By introducing hybrid integrals and polygonal shape functions, not only is the overall computational dimensionality reduced, but geometric accuracy and numerical stability are also effectively improved. Finally, by combining the Newmark method with a synchronous iterative algorithm to solve the time-history response equation, the displacement, stress, and crack propagation characteristics of hollow lightweight partition walls under different load conditions can be efficiently obtained, thereby achieving high-precision and high-efficiency fracture calculation. This invention significantly reduces the computational cost of nonlocal dynamic analysis of hollow lightweight partition walls and improves computational efficiency and fracture simulation accuracy.

[0180] Based on Example 1, this example details the verification of a method for calculating crack propagation in hollow lightweight foundation pits or dam partition walls. Numerical examples are analyzed from multiple perspectives, specifically as follows:

[0181] The free vibration characteristics of hollow lightweight partition wall panels under simply supported boundary conditions were analyzed. By comparing the results with those of traditional finite element methods, the superiority of this method in terms of geometric accuracy and frequency response prediction was verified. In the future, the mechanical response characteristics under complex conditions such as concentrated loads and crack penetration will be further investigated to comprehensively evaluate the performance of the Peri-PolySBFEM method in engineering applications.

[0182] like Figure 2 As shown, tensile tests were conducted on hollow lightweight foundation pit or dam partition walls. The hole sizes were randomly generated between 2.5 mm and 5 mm. The elastic modulus E of the wall was 30 GPa, and the critical length was 0.03 mm. The mesh size of the hollow lightweight partition wall was 1.0 mm, with 18,763 triangular meshes, 11,496 quadrilateral meshes, and 9,802 polygonal meshes. The displacement increment was 0.02 mm.

[0183] like Figure 3 As shown, Figure 3 The image shows the crack propagation results of a hollow lightweight partition wall panel under a triangular mesh. The panel area is represented by dense material points, and the white circles represent the hollow cavities. The color scale on the right ("damage") corresponds to the degree of damage (higher values ​​indicate more severe damage). It can be seen that the crack initiates in the upper part of the panel and propagates downwards in an approximately vertical direction, with the propagation path closely following the stress concentration area at the edge of the cavity. The damage concentration zone (dark area) has a uniform width, and the gradient of damage degree of the material points is smooth. This indicates that the triangular mesh accurately matches the geometric constraints of the hollow panel's multi-cavity structure when describing the stress transfer and damage evolution in the early stages of crack initiation. This avoids numerical fluctuations caused by mesh distortion and fully reproduces the morphological characteristics of the initial crack.

[0184] like Figure 4 As shown, Figure 4 The calculation results are shown under a quadrilateral mesh: the plate region is divided into quadrilateral elements, and hollow chambers are also identified by white circles. The damage scale logic is consistent with... Figure 3 Consistent. The location of the crack initiation is the same as... Figure 3The results are consistent, but the expansion path shows local fluctuations. When passing through small and medium-sized chambers, the damage zone exhibits a brief "narrowing-widening" change. The regularity of the quadrilateral units makes the distribution of material points away from the crack area more uniform. However, at geometric abrupt changes such as the corners of the chambers, the local peak of the damage degree is slightly higher than that of the triangular mesh scenario. This difference reflects the subtle limitations of the quadrilateral mesh in terms of geometric adaptability, but it can still effectively capture the macroscopic expansion direction and bifurcation trend of the crack.

[0185] like Figure 5 As shown, Figure 5 This demonstrates crack propagation within a polygonal mesh: the plate is discretized using irregular polygonal elements, and the chamber markings and damage scale are consistent. The crack initiation and propagation trajectories are shown in the diagram. Figure 3 The damage zones are highly similar, with uniform width and a more natural transition in damage intensity when passing through chambers of different sizes. The "adaptive geometric fitting" characteristic of polygonal meshes allows them to retain the smoothness of triangular meshes while having the computational efficiency of quadrilateral meshes when fitting the complex boundaries of hollow slab multi-cavity structures. The spatial relationship between crack propagation paths and chambers is more consistent with the stress fracture law of hollow slabs in actual engineering.

[0186] This embodiment details how three mesh types can meet the computational requirements for crack propagation in hollow lightweight partition walls. The core differences lie in the presentation of details and geometric adaptability: triangular and polygonal meshes perform better in terms of crack smoothness and damage gradient rationality, and are more suitable for complex geometric scenarios of multi-cavity structures; although quadrilateral meshes have slight fluctuations in local details, the macroscopic results are still reliable. This comparison not only verifies the compatibility of this method with different mesh types, but also provides an intuitive basis for engineers to select mesh types according to computational accuracy and efficiency requirements.

[0187] The above are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. For those skilled in the art, the present invention can have various modifications and variations. Any changes, modifications, substitutions, integrations, and parameter changes made to these embodiments within the spirit and principles of the present invention, without departing from the principles and spirit of the present invention, through conventional substitutions or to achieve the same function, fall within the scope of protection of the present invention.

Claims

1. A method for calculating crack propagation in hollow lightweight foundation pit or dam partition wall panels, characterized in that, include: Obtain the material parameters, geometric parameters, and boundary conditions of the hollow lightweight partition wall panel. The material parameters include elastic modulus, mass density, and Poisson's ratio. The boundary conditions include load boundaries and constraint boundaries. Based on the scaled boundary finite element method, the irregular polygonal physical domain in the Cartesian coordinate system is mapped to the standard scaled boundary coordinate system, thereby realizing the geometric normalization of the physical domain; By utilizing the separation property of radial and circumferential variables in the proportional boundary coordinate system, an isoparametric polygon shape function adapted to polygon geometry is derived to describe the potential field distribution in the physical domain. A hybrid integration scheme combining circumferential second-order Gaussian quadrature and radial semi-analytical solution is adopted to calculate the nonlocal integral of bond basis peri-field dynamics; Based on the aforementioned isoparametric polygon shape function and hybrid integral scheme, the near-field dynamic momentum equation of the discrete bond basis is used to establish the relationship between material points and element geometry. Substitute the initial global stiffness matrix and iteratively solve for the displacement of material points. Combine this with the nonlocal fracture criterion to capture the initiation of microcracks, the propagation of macrocracks, and the penetration process within the partition wall.

2. The method for calculating crack propagation in hollow lightweight foundation pit or dam partition wall panels according to claim 1, characterized in that, The physical domain geometric normalization is achieved through isoparametric mapping technology, and the coordinate interpolation formula for any point on the three-node line element is: in, For the circumferential coordinates of the proportional boundary coordinate system, The shape function of the three-node line element; any point on the three-node line element The coordinates are represented as: , ,in, , , , These are the Cartesian coordinates of the three nodes of the three-node line element; The formula for the coordinates of any point within a triangular sector is: , ,in, The radial coordinates are those of the proportional boundary coordinate system. , Let be the coordinates of any point within the triangular sector in the proportional boundary coordinate system.

3. The method for calculating crack propagation in hollow lightweight foundation pit or dam partition wall panels according to claim 2, characterized in that, The geometric transformation relationship between the Cartesian coordinate system and the proportional boundary coordinate system is derived based on the Jacobian matrix, which is: ,in, , They are respectively , right The partial derivatives, , They are respectively , right The partial derivatives, It is the Jacobian matrix of a three-node line element; , , Shape function right The partial derivatives, The determinant of the Jacobian matrix is: The mapping formula for the differential operator ∇ is: ,in, , , , This is the vector of coefficients for the differential operator mapping.

4. The method for calculating crack propagation in hollow lightweight foundation pit or dam partition wall panels according to claim 1, characterized in that, The isoparametric polygon shape function is derived based on the Laplace equation, which is: ,in, Let be the potential function at a point within a triangular sector, used to describe the potential field distribution within the physical domain; the expression for the potential function is: , It is only related to the radial coordinate Related radial analytic functions, The shape function of the three-node line element; The Euler-Cauchy differential equation is obtained by transformation using the weighted residual method: 0, where, , and These are the coefficient matrices, , They are respectively right The first and second partial derivatives, for The transpose of the matrix; , , , It is a geometric interpolation matrix, and is related to... Irrelevant , Let be the vector of coefficients mapped by the differential operator. For shape function pairs The partial derivatives of .

5. The method for calculating crack propagation in hollow lightweight foundation pit or dam partition wall panels according to claim 4, characterized in that, Transform the second-order differential equation into a first-order differential equation: ,in, It is an analytic function related to internal flux. is the Hamiltonian matrix, used to reduce the order of second-order differential equations; The expression for the Hamiltonian matrix Z is: ,in, , They are respectively , The inverse matrix; We obtain this through eigenvalue decomposition: ,in, It contains eigenvalues Transformation matrix of relevant eigenvectors, This represents the corresponding three-node line unit. Let be the integration constant, and , The eigenvector matrix, It is the potential at the node; The expression for the shape function of an isoparametric polygon is: ,in, It is an isoparametric polygon shape function adapted to polygon geometry, used to describe the continuous distribution of potential field in physical domain.

6. The method for calculating crack propagation in hollow lightweight foundation pit or dam partition wall panels according to claim 1, characterized in that, The momentum equation for the near-field dynamics of the bond group is: ,in, The mass density of the hollow lightweight partition wall panel. For material points At any moment acceleration vector, For material points The volume of the influence domain To influence any material point within the domain, , Material points , At any moment The displacement vector, For paired force functions, It is a volume force vector; The equations for the paired force vectors are: ,in, For material points and position vector difference, For material points and The difference in displacement vectors, for The model; The force function per unit volume of the prototype microelastic material is: ,in, For bond elongation, for The model, for The model, Here are the fracture parameters, and , This represents the critical bond elongation. For the micromodulus function, The radius of the influence domain.

7. The method for calculating crack propagation in hollow lightweight foundation pit or dam partition wall panels according to claim 6, characterized in that, Micromodulus function The expression is: ,in, For the initial micromodulus, The influence domain weighting function; ,in, The elastic modulus of the hollow lightweight partition wall panel. Poisson's ratio; in, for The model; time Material Point The damage variables are: ,in, As a damage variable, it is used to quantitatively characterize material points. At any moment The damage state is represented by the numerator as the volume integral of the broken bonds within the influence domain, and the denominator as the total volume of the influence domain.

8. The method for calculating crack propagation in hollow lightweight foundation pit or dam partition wall panels according to claim 1, characterized in that, The momentum equation is discretized using the Galerkin weighted residual method, and the weak integral control equation is: ,in, To solve for the domain, For the weight function, It is a volume force vector. , Material points The volume element; The approximate solution for displacement and the weighting function are as follows: , ,in, For the shape function matrix, This is a column vector of nodal displacements; The weak-form equation after discretization of the Gaussian integral is: ,in, , These represent the number of Gaussian integration points within the solution domain and the influence domain, respectively. , These are the solution domain and the influence domain, respectively. The coordinates of a Gaussian point , These are the volume weights corresponding to the Gaussian points. for The transpose of .

9. The method for calculating crack propagation in hollow lightweight foundation pit or dam partition wall panels according to claim 1, characterized in that, The matrix form of paired force vectors in the global coordinate system is: ,in, As paired force vectors, , Material points exist directional force component, , Material points exist directional force component, For fracture parameters, For micro stiffness matrix, It is a displacement vector. , Material points exist Displacement in direction, , Material points exist Displacement in direction; Microstiffness matrix in local coordinate system for: ,in, For the micromodulus function, for The model; The transformation relationship between the global and local coordinate systems is as follows: ,in, Let be the paired force vector and displacement vector in the local coordinate system, respectively, and T be the transformation matrix. , , For material points Cartesian coordinates, , For material points Cartesian coordinates.

10. The method for calculating crack propagation in hollow lightweight foundation pit or dam partition wall panels according to claim 9, characterized in that, The global microstiffness matrix is: ,in, Here is the microstiffness matrix of the local coordinate system. Transformation matrix The transpose of the matrix, , They are respectively , The square of, for and The product; The bond stiffness matrix is: ,in, For material points For material points The bond stiffness matrix, For material points Volume weight, , Material points , The shape function; The final matrix form of the momentum equation is: ,in, The global stiffness matrix. This is the global node displacement column vector. This is a global external load column vector used for iteratively solving for the displacement of material points.

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