Symmetric Galerkin Boundary Element Method Research Articles

An efficient computational framework for height-contained growing and intersecting hydraulic fracturing simulation via SGBEM–FEM

An efficient computational framework to model the growth of intersecting height-contained hydraulic fractures is proposed. The fracture in solids is modeled by the classical theory of linear elastic fracture mechanics, whereas the injection flow in hydraulic fractures is treated as channel flow. The governing equations of fracture mechanics are formulated in terms of weakly-singular weak-form boundary integral equations, with the knowledge of the fracture surface geometry exploited to reduce the integral dimension. The fluid flow within the fracture is treated using finite element method as one dimensional flow in an arbitrarily curved channel. The symmetric Galerkin boundary element method and finite element method are coupled with the traction continuity condition. The model is then formulated in a variational form and augmented with Lagrange multipliers to prescribe displacement constraints on intersections of hydraulic fractures, while fluid constraints are enforced by a consolidation of associated pressure degrees of freedom. Pertinent numerical implementation techniques, including spatial and temporal discretizations, special crack tip element, crack propagation criteria and remeshing strategy, are presented. The proposed framework is verified by comparing to the numerical results from a three dimensional hydraulic fracturing simulator and then verified by comparing the stress intensity factors to the XFEM and Murakami’s results for both star shaped cracks and symmetric branched cracks. The verification shows the utility properly enforces the physical constraints on the intersections of fractures. Three numerical examples are shown to demonstrate the capability of the proposed computational framework modeling the intersecting hydraulic fracturing process in a single or separated height-contained pay zones.

B-Spline Surface-Based Reduced-Order Modeling of Nonplanar Crack Growth in Structural Digital Twins

Nonplanar crack growth holds a critical role in aeronautical structures, necessitating effective analysis under mixed fatigue loading to assess structural integrity. This study introduces a reduced-order modeling (ROM) method for predicting nonplanar crack growth in structural digital twins. The method’s advantage lies in its representation of the entire crack surface morphology using a B-spline surface, which better captures its impact on crack growth. The symmetric Galerkin boundary element method–finite element method coupling method is adopted as a full-order method to generate the crack database. Isoparametric coordinates of the crack surface and stress intensity factor serve as input and output, respectively, for training the ROM, which integrates -means clustering, principal component analysis, and Gaussian process regression. The proposed approach is demonstrated using a rotorcraft-mast-like component. Results reveal superior fracture mechanics parameter prediction compared to the crack-front-based ROM. Furthermore, the method boasts three orders of magnitude greater efficiency than full-order simulation, enabling its coupling with approaches like Monte Carlo for probabilistic crack growth analysis. Future work entails integrating our method into the probabilistic framework of digital twins.

Shear deformable plate by SGBEM: Indirect “progenitor matrix” approach

The Mindlin-Reissner's formulation of shear deformable plate is addressed through the Symmetric Galerkin Boundary Element Method (SGBEM). The thick plate's elastic problem is resolved and fundamental solutions matrix is obtained. The application of Betti's theorem in the unlimited domain allows obtaining the Somigliana's Identities (S.I.s). The solving system is obtained by an indirect approach, i.e., through the “progenitor matrix”. The “progenitor matrix” allows simulating all possible plate's load conditions and is the starting point for an automatic calculation code. The coefficients are calculated through the theory of distributions, using double integrals without resorting to regularization techniques but by exploiting the expansion in power series of the Bessel functions present in the fundamental solutions.Particular attention is paid to the presence of domain loads, whose domain integral is transformed into a boundary one using the Radial Integral Method technique (RIM).These strategies lead to a robust procedure that allows obtaining good results even in the presence of boundary sparse discretization. This is demonstrated by the results of the examples carried out which, compared with the analytical solutions present in the literature, show a very good convergence.

Analysis of near-interface cracks in three-dimensional anisotropic multi-materials by efficient BIEM

This paper presents the full analysis of near-interface cracks in three-dimensional linear elastic multimaterial bodies by a weakly singular boundary integral equation method (BIEM). The proposed technique was implemented in a general framework that allowed the treatment of finite cracked bodies with general configurations, material anisotropy, and mode-mixity. A system of integral equations governing unknown data on the boundary, crack surface, and material interface was established using a pair of weakly singular weak-form displacement and traction integral equations together with continuity along the material interface. A symmetric Galerkin boundary element method along with the finite element technique were implemented to solve the governing integral equations. In addition, the special near-front interpolation was employed to enhance the approximation of the relative crack-face displacement in the neighborhood of the crack front. The solved crack-face data were used directly to post-process the stress intensity factors and T-stresses along the crack front. An extensive numerical study was carried out not only to demonstrate the accuracy, convergence, and capability of the proposed technique, but also to explore the influence of material stiffness and distance to the material interface on fracture data along the crack front.

Pixel-based boundary element method for computing effective thermal conductivity of heterogeneous materials

In the present article, we propose an efficient 2D Pixel-based Boundary Element formulation to compute the effective thermal conductivity of heterogeneous materials. In the proposed formulation, each pixel of the digital image is represented by a subdomain with four boundary elements. We develop the formulation within the context of the Symmetric Galerkin Boundary Element Method and, apart from the Boundary Element Method itself, the other main ingredients are the Domain Decomposition technique, to treat each pixel as a subdomain, and the Element-by-Element technique used in conjunction with the Preconditioned Conjugate Gradient method to solve the resulting symmetric positive definite large linear system. In addition, Periodic Boundary Conditions are employed. We validate our formulation and implementation with the aid of analytical results for the case of a plate with periodic circular inclusions. Further, we illustrate the application of the proposed formulation by computing the effective thermal conductivity of a plate with periodic square inclusions, a woven-like material, and a cast iron sample, and compare the obtained results with those obtained via finite elements. Our results indicate that the Pixel-based Boundary Element formulation performs, at least, as good as the Pixel-based Finite Element formulation.

Weakly-singular symmetric Galerkin boundary element method in thermoelasticity for the fracture analysis of three-dimensional solids

The Symmetric Galerkin Boundary Element Method has demonstrated significant advantages for the fracture mechanics analysis of three-dimensional solid structures. However, when dealing with structures in thermal environments, strongly-singular domain integrals resulting from the temperature field require meshing of the interior of the domain. In this paper, formulation and implementation of the weakly-singular SGBEM of three-dimensional thermoelastic intact and cracked solids are presented where temperature-induced domain integrals are transformed into boundary integrals. The strong singularity of these temperature-induced integrals is reduced to the weak singularity by exploring the feature of double-layer surface integrals in the SGBEM. Obtained temperature-induced boundary integrals affect only the right-hand-side vector of the SGBEM equation, and therefore the advantages of the SGBEM, such as the symmetry of the stiffness matrix, are retained. Several examples of three-dimensional solids with or without cracks subjected to thermal loadings are presented to validate the developed methods. Advantages and disadvantages of SGBEMs with different temperature-induced boundary integrals are also discussed.

Real-Time Prediction of Probabilistic Crack Growth with a Helicopter Component Digital Twin

To deploy the airframe digital twin or to conduct probabilistic evaluations of the remaining life of a structural component, a (near) real-time crack-growth simulation method is critical. In this paper, a reduced-order simulation approach is developed to achieve this goal by leveraging two methods. On the one hand, the symmetric Galerkin boundary element method–finite element method (SGBEM-FEM) coupling method is combined with parametric modeling to generate the database of computed stress intensity factors for cracks with various sizes/shapes in a complex structural component, by which hundreds of samples are automatically simulated within a day. On the other hand, machine learning methods are applied to establish the relation between crack sizes/shapes and crack-front stress intensity factors. By combining the reduced-order computational model with load inputs and fatigue growth laws, a real-time prediction of probabilistic crack growth in complex structures with minimum computational burden is realized. In an example of a round-robin helicopter component, even though the fatigue crack growth is simulated cycle by cycle, the simulation is faster than real-time (as compared with the physical test). The proposed approach is a key simulation technology toward realizing the digital twin of complex structures, which further requires fusion of model predictions with flight/inspection/monitoring data.

Weakly Singular Symmetric Galerkin Boundary Element Method for Fracture Analysis of Three-Dimensional Structures Considering Rotational Inertia and Gravitational Forces

The Symmetric Galerkin Boundary Element Method is advantageous for the linear elastic fracture and crackgrowth analysis of solid structures, because only boundary and crack-surface elements are needed. However, for engineering structures subjected to body forces such as rotational inertia and gravitational loads, additional domain integral terms in the Galerkin boundary integral equation will necessitate meshing of the interior of the domain. In this study, weakly-singular SGBEM for fracture analysis of three-dimensional structures considering rotational inertia and gravitational forces are developed. By using divergence theorem or alternatively the radial integration method, the domain integral terms caused by body forces are transformed into boundary integrals. And due to the weak singularity of the formulated boundary integral equations, a simple Gauss-Legendre quadrature with a few integral points is suffcient for numerically evaluating the SGBEM equations. Some numerical examples are presented to verify this approach and results are compared with benchmark solutions.

Numerical Study on the Fracture Properties of Concrete Shield Tunnel Lining Segments

Shield tunnel lining structure is usually under very complex loading conditions in the underground space. As a kind of the common concrete structures, any defect in the tunnel lining segment may deteriorate its bearing capacity and even cause severe disasters. Three-dimensional numerical models of shield tunnel lining segments with initial cracks are built using the Symmetric Galerkin Boundary Element Method- (SGBEM-) Finite Element Method (FEM) Alternating Method. The cracking load and ultimate load of the tunnel segments are obtained, and crack propagation under fatigue load is also simulated by employing the Paris Fatigue Law. Results show that loading eccentricity has a very large influence on the bearing capacity of the cracked lining segment; the larger the loading eccentricity, the smaller the bearing capacity. Deformation and damage of the lining segment show obvious phases, which consist of the initial crack, cracking stage, steady crack propagation, unsteady crack propagation, and eventual failure.

SGBEM Analysis of Diffraction of P- and SV-Waves by a Plane Crack in an Infinite Domain

Diffraction of elastic waves by a plane crack in an unbounded domain was investigated in this work using the symmetric Galerkin boundary element method (SGBEM) for elastodynamics in the frequency domain. The types of wave considered herein are the incident plane harmonic compressional wave (P-wave) and the incident plane harmonic shear wave (SV-wave). By using the scalar and vector potential functions of these incident waves, the wave displacement field and thereby, the wave traction field can be identified. To solve for the scattered field problem of these incoming waves on a crack in an infinite plate using the SGBEM, the crack faces were loaded with tractions equal and opposite to those produced at the same position by the incident waves. The scattering displacements and tractions found were then employed in a new boundary integral formula developed for computing the dynamic T-stress (DTS) for cracks in an infinite domain, while only the scattering displacements were needed for determining the dynamic stress intensity factors (DSIFs) by the displacement correlation technique. Numerous numerical examples were presented in this work. The SGBEM results for the DSIFs agreed very well with those obtained from using the integral transform method, and to the best knowledge of the authors, the SGBEM results for the DTS were presented for the first time in this work.

A dual interpolation Galerkin boundary face method for potential problems

A dual interpolation Galerkin boundary face method (DiGBFM) is applied in this paper by combining the newly developed dual interpolation method with the Galerkin boundary face method (BFM). The dual interpolation method unifies the conforming and nonconforming elements in the BFM implementation. It classifies the nodes of a conventional conforming element into virtual nodes and source nodes. Potentials and fluxes are interpolated using the continuous elements in the same way as conforming BFM, while boundary integral equations (BIEs) are collocated at source nodes, in the same way as nonconforming BFM. In order to arrive at a square linear system, we provide additional constraint equations, which are established by the moving least-squares (MLS) approximation, to condense the degrees of freedom relating to virtual nodes. Compared with the traditional symmetric Galerkin boundary element method (BEM), the symmetry feature of the DiGBFM equations is obtained simply through matrix manipulations, because of the use of the symmetric BEM, and no hypersingular BIE is needed in the DiGBFM. The proposed method has been implemented successfully for solving 2-D steady-state potential problems. Several numerical examples are presented in this paper to show the convergence and accuracy of this new method.

Removing Non-Uniqueness in Symmetric Galerkin Boundary Element Method for Elastostatic Neumann Problems and its Application to Half-Space Problems

ABSTRACTWhen the Symmetric Galerkin boundary element method (SGBEM) based on full-space elastostatic fundamental solutions is used to solve Neumann problems, the displacement solution cannot be uniquely determined because of the inevitable rigid-body-motion terms involved. Several methods that have been used to remove the non-uniqueness, including additional point support, eigen decomposition, regularization of a singular system and modified boundary integral equations, were introduced to amend SGBEM, and were verified to eliminate the rigid body motions in the solutions of full-space exterior Neumann problems. Because half-space problems are common in geotechnical engineering practice and they are usually Neumann problems, typical half-space problems were also analyzed using the amended SGBEM with a truncated free surface mesh. However, various levels of errors showed for all the methods of removing non-uniqueness investigated. Among them, the modified boundary integral equations based on the Fredholm’s theory is relatively preferable for its accurate results inside and near the loaded area, especially where the deformation varies significantly.

Forward and inverse problems in piezoelectricity using isogeometric symmetric Galerkin boundary element method and level set method

This work presents the method to combine isogeometric analysis coupled to symmetric Galerkin boundary element method (IGA-SGBEM) and parametric level set (PaLS)-based optimization scheme for the analysis of linear problems in two-dimensional piezoelectric media. IGA-SGBEM is used to obtain field variables (i.e. generalized displacement and traction) in the forward analysis. Then, inverse analysis of flaw detection in piezoelectric structures is attempted by combining IGA-SGBEM with PaLS-based optimization scheme. In this proposed method, the versatility of isogeometric analysis (IGA) is proved in the inverse progress, where the iso-line of the level set function can be easily reconstructed and incorporated into the IGA framework. Numerical examples are examined to validate and to demonstrate the robustness of the proposed method in solving both forward and inverse problems in piezoelectricity.

Optimizations of a fast multipole symmetric Galerkin boundary element method code

This paper presents some optimizations of a fast multipole symmetric Galerkin boundary element method code. Except general optimizations, the code is specially sped up for crack propagation problems. Existing useful computational results are saved and re-used during the propagation. Some time-consuming phases of the code are accelerated by a shared memory parallelization. A new sparse matrix method is designed based on coordinate format and compressed sparse row format to limit the memory required during the matrix construction phase. The remarkable performance of the new code is shown through many simulations including large-scale problems.

Dynamic stress intensity factors for multiple parallel cracks in an infinite domain under the passage of a normal incident impact or blast P-wave

A numerical technique based upon the symmetric-Galerkin boundary element method (SGBEM) for 2-D elastodynamics in the Laplace-space frequency domain and the principle of superposition of loads was introduced in this work to provide new numerical results for the dynamic stress intensity factors (DSIFs) for multiple parallel cracks in an infinite domain under the passage of an incident impact or blast P-wave normal to the crack faces. Seven test examples were presented and, compared to some DSIF results available in the literature, our Laplace-SGBEM results are able to clearly show the impact of wave scattering on the DSIFs. The results also depict a behavior known as the damped-like oscillation of the DSIFs about their corresponding static value which is caused by the interaction between scattered waves and the crack surfaces. The new results are expected to serve as benchmarks for future techniques to measure their own performance.

Analysis for T-stress of cracks in 3D anisotropic elastic media by weakly singular integral equation method

This paper proposes an efficient numerical procedure for computing T-stresses of cracks in three-dimensional, linearly elastic, infinite media. The technique is established in a broad framework allowing a medium made of generally anisotropic materials and cracks of arbitrary shape and under general loading conditions to be treated. A pair of weakly singular, weak-form, displacement and traction boundary integral equations is utilized to formulate the key equations governing the unknown crack-face fields. Besides the basic benefits such as the reduction of spatial dimensions of the solution space and the efficient treatment of unbounded domains and remote boundary data, use of such integral equations in the formulation offers additional positive features including the computational simplicity resulting from the weakly singular nature of all involved integrals and the involvement of a complete set of crack-face displacement fields. A weakly singular symmetric Galerkin boundary element method together with the special near-front approximation is utilized to solve for the unknown relative crack-face displacement whereas the sum of the crack-face displacement is obtained from the displacement boundary integral equation for cracks via the Galerkin technique. The latter step is one of the essential aspects of the present study that provides the direct means for determining the T-stress data in terms of the sum of the crack-face displacement in the neighborhood of the crack-front. An extensive numerical study is conducted for various scenarios and a selected set of results is reported to demonstrate both the accuracy and capability of the proposed technique.

Symmetric Galerkin boundary element analysis of the interaction between multiple growing cracks in infinite domains

The use of the symmetric Galerkin boundary element method (SGBEM) for studying the quasi-static interaction between multiple growing micro-cracks is presented in this work. The micro-cracks can conveniently be modeled in infinite domains, and this type of analysis can be handled by the SGBEM in a straightforward manner. In fact, it reduces the size of the analysis due to the absence of a physical boundary. A quasi-static multi-crack growth model based upon the maximum hoop stress criterion (MHSC), and the SGBEM was developed in this work. An improved quarter-point crack-tip element and adjusted maximum crack increments were employed to enhance the accuracy and effectiveness of the crack growth prediction. The improved quarter-point element has been known for producing accurate stress intensity factors required by the MHSC, while the technique used to adjust the maximum crack increment at each iteration of crack growth simulations allows to achieve converged (accurate) crack extension paths even if a relatively large maximum crack increment is selected at the onset. Several numerical examples were presented to show the effectiveness of the proposed multi-crack growth model.

Calculation of T-stress for cracks in two-dimensional anisotropic elastic media by boundary integral equation method

A numerical procedure is proposed to compute the T-stress for two-dimensional cracks in general anisotropic elastic media. T-stress is determined from the sum of crack-face displacements which are computed via an integral equation of the boundary data. To smooth out the data in order to perform accurately numerical differentiation, the sum of crack-face displacement is established in a weak-form integral equation in which the integration domain is simply the crack-tip element. This weak-form integral equation is then solved numerically using standard Galerkin approximation to obtain the nodal values of the sum of crack-face displacements. The procedure is incorporated in a weakly-singular symmetric Galerkin boundary element method in which all integral equations for the traction and displacement on the boundary of the domain and on the crack faces include (at most) weakly-singular kernels. To examine the accuracy and efficiency of the developed method, various numerical examples for cracks in infinite and finite domains are treated. It is shown that highly accurate results are obtained using relatively coarse meshes.

Interfacial debonds of layered anisotropic materials using a quasi-static interface damage model with Coulomb friction

A new quasi-static and energy based formulation of an interface damage model which includes Coulomb friction at the interface between anisotropic solids is provided. The interface traction-relative displacement response is based on an assumption of a thin adhesive layer whose behaviour is analogous to cohesive zone models. The damaged interface is considered, if exposed to a pressure, as a contact zone where Coulomb friction law is also taken into account. As the contacting solids are generally anisotropic, the friction may exhibit some anisotropic behaviour, too, which is included into the proposed model. The solution of the problem is sought numerically by a semi-implicit time-stepping procedure which uses recursive decoupled double minimisation in displacements and damage variables. The spatial discretisation is based on the symmetric Galerkin boundary-element method of a multidomain problem, where the interface variables are calculated by sequential quadratic programming, being a tool for resolving each partial minimisation in the proposed recursive scheme. Sample numerical examples demonstrate applicability of the described model.

Efficient assembly based on B-spline tailored quadrature rules for the IgA-SGBEM

This paper deals with the discrete counterpart of 2D elliptic model problems rewritten in terms of Boundary Integral Equations. The study is done within the framework of Isogeometric Analysis based on B-splines. In such a context, the problem of constructing appropriate, accurate and efficient quadrature rules for the Symmetric Galerkin Boundary Element Method is here investigated. The new integration schemes, together with row assembly and sum factorization, are used to build a more efficient strategy to derive the final linear system of equations. Key ingredients are weighted quadrature rules tailored for B-splines, that are constructed to be exact in the whole test space, also with respect to the singular kernel. Several simulations are presented and discussed, showing accurate evaluation of the involved integrals and outlining the superiority of the new approach in terms of computational cost and elapsed time with respect to the standard element-by-element assembly.