Ivo Babuška’s research while affiliated with University of Texas at Austin and other places

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Publications (397)


An enriched immersed finite element method for interface problems with nonhomogeneous jump conditions
  • Article

February 2023

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72 Reads

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36 Citations

Computer Methods in Applied Mechanics and Engineering

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Ivo Babuška

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This article presents the first higher degree immersed finite element (IFE) method with proven optimal convergence for elliptic interface problems with nonhomogeneous jump conditions. It also gives the first analysis for the condition numbers of the resulting systems including the optimal upper bounds with respect to the mesh size and its robustness with respect to small-cut interface elements. In this method, jump conditions are approximated optimally by basic IFE and enrichment IFE which are piecewise pth degree polynomial functions constructed by solving local Cauchy problems on interface elements. The proposed IFE method is based on a discontinuous Galerkin formulation on interface elements and a continuous Galerkin formulation on non-interface elements.



A 3-D body Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} containing a traction-free crack ΓO\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _O$$\end{document}; the parts of boundary marked with “///” (top and bottom) constitutes the ΓD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _D$$\end{document}, where the essential boundary condition is imposed. The natural boundary condition is imposed on the rest of the boundary ΓN=∂Ω¯\ΓD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _N = \partial \overline{\Omega }\setminus \Gamma _D$$\end{document}
The various enrichment schemes. The nodes “∙\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bullet $$\end{document}” are enriched by the Heaviside functions. The nodes “×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times $$\end{document}” are enriched by the singular functions. Left: topological enrichment; Right: geometric enrichment, where some nodes (“∙\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bullet $$\end{document} and “×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times $$\end{document}”) are enriched by both the Heaviside and singular functions
The crack ΓO\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _O$$\end{document} passes through the middle of the elements. Left: the domain. Right: the mesh
The crack ΓO\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _O$$\end{document} passes through the middle of the elements. The H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1$$\end{document} errors of topological and geometric GFEM, h.l.SGFEM. Left: the example 1. Middle: the example 2. Right: the example 3. h increases from left to right
The crack ΓO\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _O$$\end{document} passes through the middle of the elements. The SCNs of topological and geometric GFEM, h.l.SGFEM. h increases from left to right

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Stable generalized finite element method (SGFEM) for three-dimensional crack problems
  • Article
  • Full-text available

August 2022

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210 Reads

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15 Citations

Numerische Mathematik

This paper proposes a stable generalized finite element method (SGFEM) for the linear 3D elasticity problem with cracked domains. Conventional material-independent branch functions serve as singular enrichments. We prove that the proposed SGFEM with the geometric enrichment scheme yields the optimal order of convergence in the energy norm, O ( h ), for fully 3D elasticity planar crack problems; h is the mesh parameter. To improve the conditioning of SGFEM, two stability techniques have been employed, namely, (a) a cubic polynomial has been used as the PU (partition of unity), instead of the standard FE hat-functions, to address the possible almost linear dependence between the PU functions and the enrichments, and (b) a local principal component analysis (LPCA) has been implemented to address the local bad conditioning produced by multi-fold enrichments at a node. The scaled condition number for the proposed SGFEM is shown to be O(h2)O(h^{-2}) O ( h - 2 ) (same as that of a standard Finite Element Method), for various relative positions of crack surface and mesh. The robustness of the scaled condition number for the proposed SGFEM, with respect to the relative positions of the crack-surface and the element boundaries, has been observed numerically. The numerical experiments for both the planar and fully 3D planar crack problems are presented to show the efficiency of the proposed SGFEM.

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A condensed generalized finite element method (CGFEM) for interface problems

March 2022

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79 Reads

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16 Citations

Computer Methods in Applied Mechanics and Engineering

Extensive developments on various generalizations of the Finite Element Method (FEM) for the interface problems, with unfitted mesh, have been made in the last few decades. Typical generalizations and techniques include the immersed FEM, the penalized FEM, the generalized or the extended FEM (GFEM/XFEM). The GFEM/XFEM achieves good approximation by augmenting the finite element approximation space with enrichment functions, which require additional degrees of freedom (DOF). However these additional DOFs, in general, deteriorate the conditioning of the GFEM/XFEM. In this study, we propose a condensed GFEM (CGFEM) for the interface problem based on the partition of unity method and a local least square scheme. Compared to the conventional GFEM/XFEM, the CGFEM possesses the same number of DOFs as in the FEM, yields good approximation, and is well-conditioned. In comparison with the immersed FEM, the construction of shape functions of CGFEM is independent of the equation types and the material coefficients of the interface problem. As a result, the CGFEM could be applied to more general interface problems, e.g., problems with anisotropic materials and elasticity systems, in a unified approach. Moreover, the shape functions of CGFEM are continuous, and thus do not need any penalty terms and parameters. The optimal order of convergence of the CGFEM has been analyzed and established theoretically in this paper. Numerical experiments for isotropic and anisotropic interface problems and comparisons with the conventional GFEM/XFEM have also been presented to illuminate the theory and effectiveness of CGFEM.


A cut-cell finite element method for Poisson’s equation on arbitrary planar domains

September 2021

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81 Reads

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10 Citations

Computer Methods in Applied Mechanics and Engineering

This article introduces a cut-cell finite element method for Poisson’s equation on arbitrarily shaped two-dimensional domains. The equation is solved on a Cartesian axis-aligned grid of 4-node elements which intersects the boundary of the domain in a smooth but arbitrary manner. Dirichlet boundary conditions are strongly imposed by a projection method, while Neumann boundary conditions require integration over a locally discretized boundary region. Representative numerical experiments demonstrate that the proposed method is stable and attains the asymptotic convergence rates expected of the corresponding unstructured body-fitted finite element method.



Figure 1. Schematic view of the model development process.
Methodology of model development in the applied sciences

June 2021

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166 Reads

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3 Citations

Journal of Computational and Applied Mechanics

The formulation and validation of mathematical models in the applied sciences are largely consistent with the methodology of scientific research programmes (MSRP), however an essential modification is necessary: The domain of calibration has to be defined. The ranking and systematic improvement of mathematical models based on objective criteria are described and illustrated by an example. The methodology outlined in this paper provides a framework for the evolutionary development of a large class of mathematical models. Mathematical Subject Classification: 03C30, 62M86, 65N75, 65Z05



Beams, plates and shells

May 2021

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19 Reads

This chapter discusses the important class of dimensionally reduced models. The starting point is the generalized formulation of the problem of linear elasticity. In order to present the main points in a simple setting, mathematical models for beams are derived from the generalized formulation of the problem of two‐dimensional elasticity. The formulation of models for beams in three dimensions is analogous but, of course, more complicated. The formulation of plate models is analogous to the formulation of beam models. The chapter considers a restricted form of the three‐dimensional elasticity problems, using constraints and loads typically used in connection with the analysis of plates. The formulation of mathematical models for structural shells is a very large and rather complicated subject. The chapter provides a brief overview of some of the salient points.



Citations (58)


... For instance, case (1) may appear for simulating crack propagation with a background shape regular meshes [9,11,43], see Figure 1.1 for an example. Case (2) may appear when solving interface problems on a background Cartesian mesh with a fitted mesh formulation [28,29,30], while Case (3) is for an unfitted mesh formulation [2,15,23,24,38]. See Figure 1.2 for illustration. ...

Reference:

Virtual element methods based on boundary triangulation:fitted and unfitted meshes
An enriched immersed finite element method for interface problems with nonhomogeneous jump conditions
  • Citing Article
  • February 2023

Computer Methods in Applied Mechanics and Engineering

... The high precision in the SIF is achieved thanks to the good approximation by the XPIELM. The proposed XPIELMs are applied to a two dimensional (2D) Poisson crack problem [64,71], a 2D elasticity problem [5,6,72], and a fully 3D edge-crack elasticity problem [63,73,74] in a unified way, where its computational advantages are demonstrated. ...

Stable generalized finite element method (SGFEM) for three-dimensional crack problems

Numerische Mathematik

... To solve (1.1) or (1.2), one way is to use the body-fitted FEM with its mesh generated depending on the shape of the interface and the boundary of the domain [9,12]. This can be challenging especially when the interface has a complicated geometry, and more so for time-dependent problems [1,41]. There is also a large class of FEMs that do not necessitate a mesh generation that conforms to the interface, which is called unfitted FEMs. ...

A condensed generalized finite element method (CGFEM) for interface problems
  • Citing Article
  • March 2022

Computer Methods in Applied Mechanics and Engineering

... Este es el desarrollo esquemático que siguen los modelos EGDE en la actualidad. Con respecto a la predicción, "al introducir nuevos datos fuera de los lineamientos de la calibración se prueba el poder predictivo del modelo, pero si se introduce el valor de los nuevos datos aumenta la confianza de los parámetros calibrados" (Szabó y Babushka, 2021). ...

Methodology of model development in the applied sciences

Journal of Computational and Applied Mechanics

... FEM discretizes the domain into a set of overlapping meshes or "finite elements". Within each element, the solution is approximated using a polynomial basis function (Polycarpou 2022;Szabó and Babuška 2021;David Müzel et al. 2020). FVM involves dividing the domain into a set of control volumes and approximating the solution within each volume using a polynomial basis function (Ali et al. 2022;Muhammad 2021;Haider and Ahmad 2022). ...

Finite Element Analysis: Method, Verification and Validation
  • Citing Book
  • June 2021

... The numerical challenges faced in these methods are typically caused by the fact that elements are intersected by the boundaries (see Fig. 1c), leading to important simulation aspects, such as the 1. Integration of discontinuous element matrices (FCM: [5,[14][15][16][17], XFEM: [18][19][20][21], CutFEM: [22]), 2. Stabilization schemes for ill-conditioned problems (FCM: [23][24][25], XFEM: [13,26,27], CutFEM: [28,29]), 3. Implementation of weak boundary conditions on immersed boundaries (FCM: [30], XFEM: [13], CutFEM: [31]), 4. Accurate capturing of weakly or strongly discontinuous displacement fields (FCM: [32], XFEM: [33,34], Cut-FEM: [6]), 5. Mass lumping and time integration schemes for transient problems (SCM: [10,35]), to just name a few. The list given above demonstrates the multitude of important topics related to immersed boundary methods, which are entire research areas on their own. ...

A cut-cell finite element method for Poisson’s equation on arbitrary planar domains
  • Citing Article
  • September 2021

Computer Methods in Applied Mechanics and Engineering

... The stable GFEM (SGFEM) [36] solved the linear dependence between the normal and enriched bases by modifying the local approximation space under certain conditions. This was adopted, extended, and applied extensively [37][38][39][40]. Agathos et al. [41] proposed approximately orthogonal enrichment functions using Gram-Schmidt orthogonalization, which approximately eliminates the linear dependence between singular enrichment functions of the same node. ...

Strongly Stable Generalized Finite Element Method (SSGFEM) for a non-smooth interface problem II: A simplified algorithm
  • Citing Article
  • May 2020

Computer Methods in Applied Mechanics and Engineering

... There is also a large class of FEMs that do not necessitate a mesh generation that conforms to the interface, which is called unfitted FEMs. Some methods that fall into this category are the immersed FEM (IFEM) [1,20,21,31,35,37], the CutFEM [6,30], the extended FEM (XFEM) [2,32,39,40,41], and the unfitted hp method [5,10,11,38]. After fixing a mesh that is independent of the interface, the IFEM proceeds by modifying shape functions of interface elements [21]. ...

A stable generalized finite element method (SGFEM) of degree two for interface problems

Computer Methods in Applied Mechanics and Engineering

... Moreover, CEM-GMsFEMs introduce a relaxed version of the energy minimization problems to construct multiscale bases, which eliminates the necessity of solving saddle-point linear systems. Our intention here is not to present a comprehensive review of multiscale computational methods from the community, and hence, notable advancements such as heterogeneous multiscale methods [11], generalized finite element methods [2,3,32], and variational multiscale methods [19,20] are not covered. ...

Multiscale-Spectral GFEM and optimal oversampling
  • Citing Article
  • June 2020

Computer Methods in Applied Mechanics and Engineering