Global-local Enrichment Research Articles

A non-intrusive multiscale framework for 2D analysis of local features by GFEM — A thorough parameter investigation

This work comprehensively investigates key parameters associated with a recently proposed non-intrusive coupling strategy for multiscale structural problems. The IGL-GFEMgl combines the Iterative Global Local Method and the Generalized Finite Element Method with global–local enrichment, GFEMgl. Different scales of the problem are solved using distinct finite element codes: the commercial software Abaqus and a research in-house code. An Iterative Global–Local non-intrusive algorithm is employed to couple the solutions provided by the two solvers, with the process accelerated by Aitken’s relaxation. Slight modifications have been introduced, and the resulting accuracy and computational performance are discussed using numerical examples. The problems investigated explore the coupling strategy within the context of 2D linear elastic problems, which include voids and crack propagation described at the local scale solved by the in-house code. A noteworthy trade-off between reducing iterations and increasing the time to solve the local problems is observed. Despite the high accuracy achieved, the two versions of the coupling strategy, namely the monolithic and staggered algorithms, exhibit different computational performances when the GFEMgl parameters, such as the number of global–local cycles and the size of the buffer zone, are evaluated for the crack propagation simulation.

On-the-fly multiscale analysis of composite materials with a Generalized Finite Element Method

A multiscale computational framework to capture stress concentrations and localized nonlinearity in composite structures is presented. An enriched approximation space, constructed using the generalized finite element method (GFEM), is used to incorporate nonlinear, heterogeneous material behavior into coarse-scale models on the fly. Enrichment functions are constructed using the GFEM with global–local enrichment functions (GFEMgl). The auxiliary local problems associated with the GFEMgl also define fine-scale constitutive behavior that is inherited by the coarse global problem. This allows a coarse homogenized global problem to learn about material heterogeneity and/or nonlinearity on the fly, considerably increasing the flexibility of the method. On top of the explicit definition of heterogeneity in local problems, the locally defined constitutive law can incorporate further levels of heterogeneity that are not explicitly modeled at the global scale. The proposed GFEMgl comes with the efficiency and scalability characteristic of the method and greatly increases the flexibility when applied to heterogeneous structures with localized material nonlinearity.

A multiscale fracture model using peridynamic enrichment of finite elements within an adaptive partition of unity: Experimental validation

Partition of unity methods (PUM) are of domain decomposition type and provide the opportunity for multiscale and multiphysics numerical modeling. Here, we apply Peridynamic (PD) enrichment to propagate cracks in the PUM global–local enrichment scheme. We apply linear elasticity globally and PD over local zones where fractures occur. The elastic fields provide appropriate boundary data for local PD simulations on a subdomain containing the crack tip to grow the crack. Once the updated crack path is found the elastic field in the body and surrounding the crack is updated using the PUM basis with an elastic field enrichment near the crack. The subdomain for the PD simulation is chosen to include the current crack tip as well as features that influence crack growth. This paper is part II of this series and validates the combined PD/PUM simulator against the experimental results. The results of numerical simulation show that we attain good agreement between experiment and simulation with a local PD subdomain that is moving with the crack tip and adaptively chosen size.

An adaptive global–local generalized FEM for multiscale advection–diffusion problems

This paper develops an adaptive algorithm for the Generalized Finite Element Method with global–local enrichment (GFEMgl) for transient multiscale PDEs. The adaptive algorithm detects a subset of global nodes with trivial enrichments, which are exactly or close to linearly dependent from the underlying coarse FEM basis, at each time step, and then removes them from the global system. It is based on the calculation of the ratio between the largest and smallest singular values of small sub-matrices extracted from the global system of equations which introduces little overhead over the non-adaptive GFEMgl for transient PDEs. Compared to existing adaptive multiscale approaches, where either an a-posterior error estimate, a change in physical quantities, or a local problem residual is calculated, the proposed approach provides an innovative framework based on singular values. The proposed approach is shown to be robust for solving advection–diffusion problems that require detecting initial conditions with spikes and capturing moving/morphing/merging mass plumes in heterogeneous media and non-uniform flow with sharp fronts. Specific examples are provided for applications in groundwater contaminant and heat dissipation. The accuracy of the proposed adaptive GFEMgl closely matches reference fine-scale FEM solutions in the L∞ norm.

A transient global-local generalized FEM for parabolic and hyperbolic PDEs with multi-space/time scales

This paper presents a novel Generalized Finite Element Method with global-local enrichment (GFEMgl) to solve time-dependent parabolic and hyperbolic problems with subscale features in both space and time. Compared with currently available GFEMgl, the proposed method requires a solution of local problems to solve transient PDEs, possibly with a different time integrator from the global problem or other local problems. The fine-scale solution can be easily retrieved. The proposed method is tested for the solution of the heat, advection, and advection-diffusion equations whose accuracy, stability, scalability, and convergence are thoroughly analyzed. Compared to direct analysis, numerical results show that the proposed method has the following properties 1) the accuracy closely matches direct analysis result with a fine mesh; 2) critical time step size is loosened; 3) optimal convergence rate is achieved. Moreover, the proposed method allows local problems to be turned off if their solutions are not helpful for the current global time step, which can further improve its efficiency.

A fracture multiscale model for peridynamic enrichment within the partition of unity method

Partition of unity methods are of domain decomposition type and provide the opportunity for multiscale and multiphysics numerical modeling. Different physical models can exist within a partition of unity method scheme for handling problems with zones of linear elasticity and zones where fractures occur. Here, the peridynamic model is used in regions of fracture and smooth partition of unity methods is used in the surrounding linear elastic media. Our method is novel in that we evolve the crack path using peridynamics and apply the partition of unity method to compute the elastic fields in the neighborhood of the crack tip. Earlier work uses the peridynamic fields, e.g displacement, at the crack tip to approximate enrichment functions for the partition of unity methods. The method is a so-called global–local enrichment strategy. The elastic fields of the undamaged media provide appropriate boundary data for the localized peridynamic simulations. The geometry of the crack path in the damaged media is transferred to partition of unity method. Here, Heaviside and Westergaard functions are used to model the crack. We do not transfer information of the peridynamic fields to the partition of unity method, solely the crack geometry. The first steps for a combined peridynamic and partition of unity method simulator are presented. We show that the local peridynamic approximation can be utilized to enrich the global partition of unity method approximation to capture the true material response with high accuracy efficiently. Test problems are provided demonstrating the validity and potential of this numerical approach.

Non-intrusive coupling of a 3-D Generalized Finite Element Method and Abaqus for the multiscale analysis of localized defects and structural features

This paper presents a multiscale computational framework able to resolve localized defects and features such as cracks and welds in large structures. It couples Abaqus models and 3-D Generalized Finite Element (GFEM) discretizations enriched with numerically-defined functions – the GFEM with global-local enrichments (GFEMgl). The structural-scale problem is modeled in Abaqus using a coarse, 3-D mesh, suitable for capturing the global response of the structure. The GFEMgl is used to accurately model localized features of interest that are otherwise ignored by Abaqus models. The GFEMgl utilizes enrichment functions provided by the solution of, potentially multiple, local problems solved in parallel. The coupling between Abaqus and GFEMgl models is handled by the Iterative Global-Local method (IGL). The proposed multiscale framework – denoted by the acronym IGL-GFEMgl in this work – is non-intrusive in the sense that the interactions between Abaqus and the GFEMgl solver require no modifications of Abaqus or knowledge about its implementation of the FEM. Numerical examples of a specimen undergoing localized plasticity, a hat-stiffened panel with a large number of spot welds, and a T-joint structure subjected to mixed-mode fatigue crack propagation are presented to demonstrate the accuracy and applicability of the proposed framework. The results show that the IGL-GFEMgl can achieve nearly the same accuracy as a Direct Finite Element Analysis (DFEA) while leveraging methodologies and algorithms implemented in commercial and research software. Moreover, it is also shown that the user time spent on model preparation can be greatly reduced when dealing with complex problems.

Stable Generalized/eXtended Finite Element Method with global–local enrichment for material nonlinear analysis

This paper presents a novel methodology for the solution of nonlinear problems using the Stable Generalized/eXtended Finite Element Method (SGFEM) in two-scale approach. The nonlinear analysis is performed in a coarse mesh and the solution of a local problem, represented by a fine mesh, is used to enrich the partition of unity of the coarse mesh in the so-called global–local strategy. The SGFEM is adopted to deal with the problem of ill-conditioning of the stiffness matrix triggered by the enrichment strategy. Here, this problem is investigated for the first time for global–local analysis of physically nonlinear problem. In addition to the use of SGFEM, it is noteworthy that the relationship between the incremental-iterative process and changes in the constitutive variables are simultaneously updated on both scales. Differently from original GFEM global–local strategies, the evolution of the state variables is conduced only in the global scale. The local problem heritages the material degradation of the global scale, by using a mapping technique, and provides the numerically obtained functions that enriches the global approximation. Different constitutive models and material laws are adopted to incorporate the material degradation. Numerical examples are presented for validating this approach and to show its generality and efficiency.

2-D Crack propagation analysis using stable generalized finite element method with global-local enrichments

In this paper, the technique of the Stable Generalized Finite Element Method (SGFEM) is applied to the numerically constructed functions of the Generalized Finite Element Method with Global-Local Enrichments (GFEMgl). The application of the resulting approach, named SGFEMgl, is expanded here to 2-D quasi-static crack propagation problems. Crack growth is performed by a two-scale strategy, using local problems generated at each propagation step – whose solutions enrich a single global problem defined on a coarse mesh. Stress Intensity Factors (SIFs) computed along crack growth, strain energy measures, performance in blending elements and the condition number are used to study the accuracy and conditioning of SGFEMgl. The method is compared with the standard GFEMgl. Numerical experiments demonstrate remarkable accuracy of SGFEMgl in linear elastic fracture mechanics problems, considering crack opening modes I and II. Convergence rates analyses also show the superiority of the method, especially with the use of geometrical enrichments.

A new approach for physically nonlinear analysis of continuum damage mechanics problems using the generalized/extended finite element method with global-local enrichment

The Generalized/Extended Finite Element Method (G/XFEM) is a numerical technique suitable to solve a wide range of continuum mechanics problems. It applies hierarchical nodal enrichment strategies within the Finite Element Method framework, allowing more flexibility in the numerical solution of boundary value problems (BVP) and being a powerful mesh-reduced method that takes advantage of the state-of-art of the standard FEM. One of the enrichment mechanisms is the so-called global-local enrichment (G/XFEMGL), in which the solution of a refined local BVP generates enrichment functions for the global domain. In this context, this work presents a new two-scale nonlinear strategy, associated with the G/XFEMGL, able to solve the material nonlinear analysis of continuum damage mechanics problems, in which quasi-brittle media degrade and soften, using a versatile mesh refinement strategy. A comprehensive description of the proposed strategy is registered and the aforementioned computational technique is validated throughout a set of plane stress numerical experiments that employ smeared crack constitutive model formulations, with well-known stress-strain laws. The results indicate that the implemented methodology could be used to solve a reasonable range of problems with considerable flexibility.

A two-scale generalized FEM for the evaluation of stress intensity factors at spot welds subjected to thermomechanical loads

This paper presents a two-scale Generalized Finite Element Method with global-local enrichments (GFEMgl) for the evaluation of stress intensity factors (SIFs) at spot welds subjected to thermomechanical loads. The method uses the solution of local boundary value problems as enrichment functions for the global/structural model and can provide accurate solutions on coarse meshes adopted at the structural scale. The faying surface and spot welds are represented by special enrichment functions instead of meshes fitting the weld geometry. This greatly simplifies the model generation process in comparison with a Direct Finite Element Analysis DFEA. Numerical experiments show that the proposed GFEMgl is able to capture the strong gradients and singularities near the spot weld edge. It can provide accurate solutions and SIFs comparable to a 3-D DFEA while requiring significantly less computational resources and user time. The robustness of the method is demonstrated through examples involving spot welds with irregular shapes and built-up panels subjected to sharp thermal fluxes.

Fracture analysis in plane structures with the two-scale G/XFEM method

Generalized or extended finite element method (G/XFEM) uses enrichment functions that holds a priori knowledge about the problem solution. These enrichment functions are mostly limited to two-dimensional problems. A well-established solution for problems without any specific types of analytically derived enrichment functions is to use numerically-build functions in which they are called global-local enrichment functions. These functions are extracted from the solution of boundary value problems defined around the region of interest discretized by a fine mesh. Such solution is used to enrich the global solution space through the partition of unity framework of the G/XFEM. Here it is presented a two-scale/global-local G/XFEM approach to model crack propagation in plane stress/strain and Reissner–Mindlin plate problems. Discontinuous functions along with the asymptotic crack-tip displacement fields are used to represent the crack without explicitly represent its geometries. Under the linear elastic fracture mechanics approach, the stress intensity factor (obtained from a domain-based interaction energy integral) can be used to either determine the crack propagation direction or propagation status, i.e., the crack can start to propagate or not. The proposed approach is presented in detail and validated by solving several linear elastic fracture mechanics problems for both plane stress/strain and Reissner–Mindlin plate cases to demonstrate its the robustness and accuracy.

Numerical analysis of a main crack interactions with micro-defects/inhomogeneities using two-scale generalized/extended finite element method

Generalized or extended finite element method (G/XFEM) models the crack by enriching functions of partition of unity type with discontinuous functions that represent well the physical behavior of the problem. However, this enrichment functions are not available for all problem types. Thus, one can use numerically-built (global-local) enrichment functions to have a better approximate procedure. This paper investigates the effects of micro-defects/inhomogeneities on a main crack behavior by modeling the micro-defects/inhomogeneities in the local problem using a two-scale G/XFEM. The global-local enrichment functions are influenced by the micro-defects/inhomogeneities from the local problem and thus change the approximate solution of the global problem with the main crack. This approach is presented in detail by solving three different linear elastic fracture mechanics problems for different cases: two plane stress and a Reissner–Mindlin plate problems. The numerical results obtained with the two-scale G/XFEM are compared with the reference solutions from the analytical, numerical solution using standard G/XFEM method and ABAQUS as well, and from the literature.

A new generalized finite element method for two-scale simulations of propagating cohesive fractures in 3-D

Summary This paper presents a novel numerical framework based on the generalized finite element method with global–local enrichments (GFEMgl) for two-scale simulations of propagating fractures in three dimensions. A non-linear cohesive law is adopted to capture objectively the dissipated energy during the process of material degradation without the need of adaptive remeshing at the macro scale or artificial regularization parameters. The cohesive crack is capable of propagating through the interior of finite elements in virtue of the partition of unity concept provided by the generalized/extended finite element method, and thus eliminating the need of interfacial surface elements to represent the geometry of discontinuities and the requirement of finite element meshes fitting the cohesive crack surface. The proposed method employs fine-scale solutions of non-linear local boundary-value problems extracted from the original global problem in order to not only construct scale-bridging enrichment functions but also to identify damaged states in the global problem, thus enabling accurate global solutions on coarse meshes. This is in contrast with the available GFEMgl in which the local solution field contributes only to the kinematic description of global solutions. The robustness, efficiency, and accuracy of this approach are demonstrated by results obtained from representative numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.

Generalized finite element approaches for analysis of localized thermo‐structural effects

SummaryThis work addresses computational modeling challenges associated with structures subjected to sharp, local heating, where complex temperature gradients in the materials cause three‐dimensional, localized, intense stress and strain variation. Because of the nature of the applied loadings, multiphysics analysis is necessary to accurately predict thermal and mechanical responses. Moreover, bridging spatial scales between localized heating and global responses of the structure is nontrivial. A large global structural model may be necessary to represent detailed geometry alone, and to capture local effects, the traditional approach of pre‐designing a mesh requires careful manual effort. These issues often lead to cumbersome and expensive global models for this class of problems. To address them, the authors introduce a generalized FEM (GFEM) approach for analyzing three‐dimensional solid, coupled physics problems exhibiting localized heating and corresponding thermomechanical effects. The capabilities of traditional hp‐adaptive FEM or GFEM as well as the GFEM with global–local enrichment functions are extended to one‐way coupled thermo‐structural problems, providing meshing flexibility at local and global scales while remaining competitive with traditional approaches. The methods are demonstrated on several example problems with localized thermal and mechanical solution features, and accuracy and (parallel) computational efficiency relative to traditional direct modeling approaches are discussed. Copyright © 2015 John Wiley & Sons, Ltd.

Generalized finite element analysis using the preconditioned conjugate gradient method

This paper introduces the generalized finite element method with global–local enrichment functions using the preconditioned conjugate gradient method. The proposed methodology is able to generate enrichment functions for problems where limited a priori knowledge on the solution is available and to utilize a preconditioner and initial guess of high quality with an addition of only small computational cost. Thus, it is very effective to analyze problems where a complex behavior is locally exhibited. Several numerical experiments are performed to confirm its effectiveness and show that it is computationally more efficient than the analysis utilizing direct methods such as the LU and Cholesky factorization methods.

Analysis and improvements of global–local enrichments for the Generalized Finite Element Method

Partition of unity methods like the GFEM, rely on closed-form analytical enrichment functions. However, only limited a priori knowledge about the solution is available for many problems of engineering relevance. Control of discretization errors in this class of problems requires mesh refinement which offsets some of the advantages associated with these methods. This has led to the development of the GFEM with global–local enrichments (GFEMgl), which uses ideas of the classical global–local finite element method to numerically build enrichment functions for the GFEM. It involves the solution of local boundary value problems using boundary conditions from the solution of the global problem discretized with a coarse mesh. These local solutions are in turn used to enrich the solution space of the global problem with the help of the partition of unity framework.This paper presents an a priori error estimate accounting for the effect of inexact boundary conditions applied to local problems. Two strategies to control the error of the GFEMgl solution due to these boundary conditions are also presented. The first one is based on the use of a buffer or over-sampling zone in the local problems. The second strategy investigated is based on multiple global–local iterations. Several numerical experiments performed on three-dimensional fracture mechanics problems illustrate the effectiveness of these strategies in controlling the error of the GFEMgl solution. The problems are selected such that the boundary conditions on the local problems range from smooth to singular functions.

Efficient analysis of transient heat transfer problems exhibiting sharp thermal gradients

In this paper, heat transfer problems with sharp spatial gradients are analyzed using the Generalized Finite Element Method with global-local enrichment functions (GFEM gl). With this approach, scale-bridging enrichment functions are generated on the fly, providing specially-tailored enrichment functions for the problem to be analyzed with no a-priori knowledge of the exact solution. In this work, a decomposition of the linear system of equations is formulated for both steady-state and transient heat transfer problems, allowing for a much more computationally efficient analysis of the problems of interest. With this algorithm, only a small portion of the global system of equations, i.e., the hierarchically added enrichments, need to be re-computed for each loading configuration or time-step. Numerical studies confirm that the condensation scheme does not impact the solution quality, while allowing for more computationally efficient simulations when large problems are considered. We also extend the GFEM gl to allow for the use of hexahedral elements in the global domain, while still using tetrahedral elements in the local domain, to allow for automatic localized mesh refinement without the use of constrained approximations. Simulations are run with the use of linear and quadratic hexahedral and tetrahedral elements in the global domain. Convergence studies indicate that the use of a different partition of unity (PoU) in the global (hexahedral elements) and local (tetrahedral elements) domains does not adversely impact the solution quality.

Analysis of three-dimensional fracture mechanics problems: A non-intrusive approach using a generalized finite element method

This paper shows that the generalized finite element method with global–local enrichment functions (GFEMgl) can be implemented non-intrusively in existing closed-source FEM software as an add-on module. The GFEMgl is based on the solution of interdependent global (structural) and local (crack) scale problems. In the approach presented here, an initial global scale problem is solved by a commercial finite element analysis software, local problems containing 3-D fractures are solved by an hp-adaptive GFEM software and an enriched global scale problem is solved by a combination of the FEM and GFEM softwares. The interactions between the solvers are limited to the exchange of load and solution vectors and does not require the introduction of user subroutines to existing FEM software. As a results, the user can benefit from built-in features of available commercial grade FEM software while adding the benefits of the GFEM for this class of problems. Several three-dimensional fracture mechanics problems aimed at investigating the applicability and accuracy of the proposed two-solver methodology are presented.

A generalized finite element method with global-local enrichment functions for confined plasticity problems

The main feature of partition of unity methods such as the generalized or extended finite element method is their ability of utilizing a priori knowledge about the solution of a problem in the form of enrichment functions. However, analytical derivation of enrichment functions with good approximation properties is mostly limited to two-dimensional linear problems. This paper presents a procedure to numerically generate proper enrichment functions for three-dimensional problems with confined plasticity where plastic evolution is gradual. This procedure involves the solution of boundary value problems around local regions exhibiting nonlinear behavior and the enrichment of the global solution space with the local solutions through the partition of unity method framework. This approach can produce accurate nonlinear solutions with a reduced computational cost compared to standard finite element methods since computationally intensive nonlinear iterations can be performed on coarse global meshes after the creation of enrichment functions properly describing localized nonlinear behavior. Several three-dimensional nonlinear problems based on the rate-independent J 2 plasticity theory with isotropic hardening are solved using the proposed procedure to demonstrate its robustness, accuracy and computational efficiency.