#### Prof. Dr.-Ing. Carolin Birk

Statik und Dynamik der Flächentragwerke

University of Duisburg-Essen

##### Contact

- carolin[dot]birk[at]uni-due[dot]de
- +49 201 183 2667
- personal website

##### Hub

**Adaptive modelling of dynamic brittle fracture - a combined phase field regularized cohesive zone model and scaled boundary finite element approach**

Natarajan, S. and Ooi, E.T. and Birk, C. and Song, C.*International Journal of Fracture*236 (2022)view abstract 10.1007/s10704-022-00634-2 **An efficient numerical scheme for the FE-approximation of magnetic stray fields in infinite domains**

Schröder, J. and Reichel, M. and Birk, C.*Computational Mechanics*70 (2022)view abstract 10.1007/s00466-022-02162-1 **Magnetostatic simulations with consideration of exterior domains using the scaled boundary finite element method**

Birk, C. and Reichel, M. and Schröder, J.*Computer Methods in Applied Mechanics and Engineering*399 (2022)view abstract 10.1016/j.cma.2022.115362 **Thermoelastic fracture analysis of functionally graded materials using the scaled boundary finite element method**

Iqbal, M.D. and Birk, C. and Ooi, E.T. and Pramod, A.L.N. and Natarajan, S. and Gravenkamp, H. and Song, C.*Engineering Fracture Mechanics*264 (2022)The scaled boundary finite element method is extended to model fracture in functionally graded materials (FGM) under coupled thermo-mechanical loads. The governing equations of coupled thermo-mechanical equilibrium are discretized using scaled boundary shape functions enriched with the thermal load terms. The material gradient is modeled as a series of power functions, and the stiffness matrix is calculated semi-analytically. Stress intensity factors and T−stress are directly calculated from their definition without any need for additional post-processing techniques. Arbitrary-sided polygon elements are employed for flexible mesh generation. Several numerical examples for isotropic and orthotropic FGMs are presented to validate the proposed technique. © 2022 Elsevier Ltdview abstract 10.1016/j.engfracmech.2022.108305 **Three-dimensional phase-field modeling of brittle fracture using an adaptive octree-based scaled boundary finite element approach**

Assaf, R. and Birk, C. and Natarajan, S. and Gravenkamp, H.*Computer Methods in Applied Mechanics and Engineering*399 (2022)view abstract 10.1016/j.cma.2022.115364 **Development of the scaled boundary finite element method for crack propagation modeling of elastic solids subjected to coupled thermo-mechanical loads**

Iqbal, M.D. and Birk, C. and Ooi, E.T. and Gravenkamp, H.*Computer Methods in Applied Mechanics and Engineering*387 (2021)This study presents the development of the scaled boundary finite element method to model discrete crack propagation induced by thermal loads. The SBFEM excels in modeling stress singularities at sharp crack tips with high accuracy. Polygon meshes are used so that a robust local re-meshing algorithm can be utilized to propagate the crack. The scaled boundary finite element formulation for steady-state thermal stress analysis is presented. Following a scaled boundary finite element analysis of a given thermal problem, the effect of initial strains due to temperature is taken into account semi-analytically in a subsequent stress analysis. Several numerical examples are presented to validate the technique and illustrate its salient features. © 2021view abstract 10.1016/j.cma.2021.114106 **A polygon scaled boundary finite element formulation for transient coupled thermoelastic fracture problems**

Ooi, E.T. and Iqbal, M.D. and Birk, C. and Natarajan, S. and Ooi, E.H. and Song, C.*Engineering Fracture Mechanics*240 (2020)The scaled boundary finite element method is developed for transient thermoelastic fracture analysis. To enable this, a set of novel shape functions are derived considering thermoelastic equilibrium. The salient features of the proposed framework are: (a) can be formulated on polygons with an arbitrary number of sides leading to flexible mesh generation and (b) facilitates an accurate and direct evaluation of the stress intensity factors from their definition without resorting to any post-processing techniques using relatively coarse meshes. Several numerical benchmark problems demonstrate the aforementioned features of the technique. © 2020 Elsevier Ltdview abstract 10.1016/j.engfracmech.2020.107300 **A scaled boundary finite element approach for shell analysis**

Wallner, M. and Birk, C. and Gravenkamp, H.*Computer Methods in Applied Mechanics and Engineering*361 (2020)A new approach to model shell structures is proposed. It is based on the scaled boundary finite element method in three dimensions. Thus, the solution is sought analytically in the through-thickness direction while the surface of the domain is discretized in a finite element sense. Since no kinematic assumptions are made, the proposed method can be applied to thick spherical shells and thin shells. Very good agreement with reference solutions is demonstrated for classical benchmark problems of shell analyses. No membrane locking induced by mesh distortion is observed. The potential of the proposed method is particularly evident when p-refinement is employed. Furthermore, the applicability of the proposed method to shells with non-spherical geometry is discussed in detail. © 2020 Elsevier B.V.view abstract 10.1016/j.cma.2019.112807 **A high-order finite element technique with automatic treatment of stress singularities by semi-analytical enrichment**

Bulling, J. and Gravenkamp, H. and Birk, C.*Computer Methods in Applied Mechanics and Engineering*355 (2019)This paper presents an approach to the automatic enrichment of finite elements in the vicinity of a stress singularity. The enrichment consists of semi-analytical singular modes constructed using the Scaled Boundary Finite Element Method (SBFEM). In contrast to analytical methods, the SBFEM provides modes for inhomogeneous and anisotropic materials without additional effort. The finite element basis can be of arbitrary order and remains unaltered by the enrichment. The approach requires enrichment in only one layer of elements around a node. Due to the compatibility of SBFEM with FEM, there is no need for transitional elements, and there are no parasitic terms. The approach is tested for several benchmark problems. The stress intensity factors are computed based on techniques inspired by the SBFEM. The proposed procedure is compared to a standard finite element implementation and shows a significant improvement in the error of the displacement field for problems involving singular stresses. © 2019view abstract 10.1016/j.cma.2019.06.025 **A scaled boundary finite element method for modelling wing crack propagation problems**

Zhang, P. and Du, C. and Birk, C. and Zhao, W.*Engineering Fracture Mechanics*216 (2019)In this paper, a new shape function based on the scaled boundary finite element method (SBFEM) with side-face loading is used to study the problem of wing crack propagation. Crack contact is modelled by introducing the contact interface constraint condition by the Lagrange multiplier method. In the crack propagation process, the contact state of the crack surface at each step is obtained by an iterative method of determination and validation, and an accurate simulation for the frictional contact expansion process of compressed wing cracks is carried out. Polygon SBFEM remeshing technology is used to simulate the propagation of single and multiple wing cracks. The correctness and effectiveness of the new SBFEM shape function method in modelling wing crack propagation are verified using related experimental and numerical simulation results. The results indicate that under pressure, the wing crack propagates along the direction of the maximum circumferential stress during initial expansion, and the crack propagation angle is −70°32′, which is a pure type II crack. The crack then opens and gradually expands towards the pressure. In the considered wing crack problems, contact phenomena only occur on the initial crack surface during crack propagation. The contact force gradually decreases from the centre to the tips of the initial crack. As the crack expands, the contact force decreases, and the contact area of the crack gradually decreases. The friction coefficient has a strong influence on the crack propagation direction. The larger the friction coefficient, the closer the propagation direction is to the linear extension in the direction of the applied load. © 2019 Elsevier Ltdview abstract 10.1016/j.engfracmech.2019.04.040 **An automatic approach for the acoustic analysis of three-dimensional bounded and unbounded domains by scaled boundary finite element method**

Liu, L. and Zhang, J. and Song, C. and Birk, C. and Gao, W.*International Journal of Mechanical Sciences*151 (2019)An automatic approach for modeling acoustic responses of 3D bounded and unbounded domains is proposed based on the scaled boundary finite element method (SBFEM). The bounded acoustic near field is modeled by SBFEM. Due to the fact that SBFEM only requires boundary discretizations, the meshing process in this approach is automatically accomplished by introducing an octree-based meshing technique and boundary trimming. Therefore, models with very complex geometries can be analyzed automatically, i.e. no human input/intervention is needed/required. For modeling the unbounded acoustic far field, a spherical high-order doubly-asymptotic open boundary is developed for modeling scalar wave propagation in 3D unbounded domain. This doubly-asymptotic open boundary is able to represent the unbounded domain accurately and efficiently as it includes recursive continued-fraction expansions in both high and low frequency limits. The continued-fraction solutions are expressed in the time-domain by introducing auxiliary variables. The bounded and unbounded domains are coupled via the nodal fluxes on the near/far field interface. The final coupled ordinary differential equations can be solve using standard time-stepping methods directly for obtaining transient acoustic responses. Various numerical examples are presented in this paper for demonstrating the accuracy, efficiency and capability of this approach for analyzing acoustic problems with complex geometries. © 2018 Elsevier Ltdview abstract 10.1016/j.ijmecsci.2018.12.018 **Automatic three-dimensional acoustic-structure interaction analysis using the scaled boundary finite element method**

Liu, L. and Zhang, J. and Song, C. and Birk, C. and Saputra, A.A. and Gao, W.*Journal of Computational Physics*395 (2019)An automatic approach for 3D analysis of acoustic-structure interaction problems is proposed based on the scaled boundary finite element method (SBFEM). The acoustic domain studied in this paper is assumed to be infinite. The infinite acoustic domain is divided into a near field (bounded domain) and a far field (unbounded domain). The acoustic near field contains structures of arbitrary shape, while the far field represents the unbounded acoustic domain. For modeling the wave propagation accurately and efficiently, continued fractions are employed to evaluate the dynamic stiffness and impedance of subdomains in both structural and acoustic domains. The time-domain equations for both structural and acoustic domains can be obtained by introducing auxiliary variables. Via satisfying the boundary conditions on the acoustic-structure interface, the global system of equations for acoustic-structure interaction system can be constructed. Symmetric formulations can also be obtained for this coupled system. Since the SBFEM requires the discretization of only the boundary, the mesh transition on the acoustic-structure interface is easily addressed by the subdivisions of 2D surface elements. Automatic meshing techniques can be incorporated in the proposed approach to generate meshes directly from the input geometrical models. Numerical examples are presented to demonstrate the accuracy, efficiency and potential of the proposed approach for modeling complex 3D acoustic-structure interaction problems. © 2019 Elsevier Inc.view abstract 10.1016/j.jcp.2019.06.033 **Dynamic soil-structure interaction in a 3D layered medium treated by coupling a semi-analytical axisymmetric far field formulation and a 3D finite element model**

Aslmand, M. and Kani, I.M. and Birk, C. and Gravenkamp, H. and Krome, F. and Eskandari-Ghadi, M.*Soil Dynamics and Earthquake Engineering*115 (2018)A strategy for dynamic soil-structure interaction problems involving three-dimensional layered soil is proposed. It is based on coupling an axisymmetric formulation for the regular layered far field region with a general three-dimensional finite element model of the near field via a cylindrical near field / far field interface. Using a virtual work statement, an axisymmetric scaled boundary finite element method (SBFEM) for the elastodynamic analysis of 3D layered continua is derived. Here, a vertical line, which is rotated around the origin of the cylindrical coordinate system is discretized in the finite element sense. A nonlinear differential equation in dynamic stiffness is obtained for each term of the Fourier series describing the circumferential variation of displacements in the far field, which must be solved numerically for each frequency of interest. While this leads to considerable savings in numerical cost compared to the effort associated with calculating a fully coupled stiffness matrix for the 3D problem, the numerical efficiency can be further increased by re-casting the problem using the method of weighted residuals. In doing so, a link with the well-known thin-layer method is established. The latter leads to a standard eigenvalue problem for the calculation of wave numbers and modes. The axisymmetric stiffness formulation obtained using one of the two aforementioned techniques is coupled with the 3D finite element model of the near field via the dynamic stiffness matrix relating nodal forces to nodal displacements at the cylindrical interface. Following a physically motivated approach, individual columns of the latter are obtained by considering situations where a unit displacement is assumed for the corresponding degree of freedom and zero displacements elsewhere and by expanding the corresponding displacement field into a Fourier series. The proposed strategy is applied to various soil-structure interaction problems involving flexible foundations of irregular shape resting on layered ground over rigid bedrock. © 2018 Elsevier Ltdview abstract 10.1016/j.soildyn.2018.07.044 **Automatic image-based stress analysis by the scaled boundary finite element method**

Saputra, A. and Talebi, H. and Tran, D. and Birk, C. and Song, C.*International Journal for Numerical Methods in Engineering*109 (2017)Digital imaging technologies such as X-ray scans and ultrasound provide a convenient and non-invasive way to capture high-resolution images. The colour intensity of digital images provides information on the geometrical features and material distribution which can be utilised for stress analysis. The proposed approach employs an automatic and robust algorithm to generate quadtree (2D) or octree (3D) meshes from digital images. The use of polygonal elements (2D) or polyhedral elements (3D) constructed by the scaled boundary finite element method avoids the issue of hanging nodes (mesh incompatibility) commonly encountered by finite elements on quadtree or octree meshes. The computational effort is reduced by considering the small number of cell patterns occurring in a quadtree or an octree mesh. Examples with analytical solutions in 2D and 3D are provided to show the validity of the approach. Other examples including the analysis of 2D and 3D microstructures of concrete specimens as well as of a domain containing multiple spherical holes are presented to demonstrate the versatility and the simplicity of the proposed technique. Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2016 John Wiley & Sons, Ltd.view abstract 10.1002/nme.5304 **Efficient wave propagation simulation on quadtree meshes using SBFEM with reduced modal basis**

Gravenkamp, H. and Saputra, A.A. and Song, C. and Birk, C.*International Journal for Numerical Methods in Engineering*110 (2017)We apply a combination of the transient scaled boundary finite element method (SBFEM) and quadtree-based discretization to model dynamic problems at high frequencies. We demonstrate that the current formulation of the SBFEM for dynamics tends to require more degrees of freedom than a corresponding spectral element discretization when dealing with smooth problems on regular domains. Thus, we improve the efficiency of the SBFEM by proposing a novel approach to reduce the number of auxiliary variables for transient analyses. Based on this improved SBFEM, we present a modified meshing procedure, which creates a quadtree mesh purely based on the geometry and allows arbitrary sizes and orders of elements, as well as an arbitrary number of different materials. The discretization of each subdomain is created automatically based on material parameters and the highest frequency of interest. The transition between regions of different properties is straightforward when using the SBFEM. The proposed approach is applied to image-based analysis with a particular focus on geological models. © 2016 John Wiley & Sons, Ltd.view abstract 10.1002/nme.5445 **Prismatic semi-analytical elements for the simulation of linear elastic problems in structures with piecewise uniform cross section**

Krome, F. and Gravenkamp, H. and Birk, C.*Computers and Structures*192 (2017)This work addresses the computation of stiffness matrices for general prismatic structures with an arbitrary cross section. The presented approach is based on the scaled boundary finite element method (SBFEM), a semi-analytical method, which can be used to model structures by only discretizing the boundary of a domain. For prismatic structures, the process is further simplified, as only the cross section of the structure has to be discretized. Thus, a particular semi-analytical finite element is constructed for bounded and unbounded domains. The proposed approach leads to a frequency-dependent stiffness matrix. This stiffness matrix can easily be coupled to other prismatic SBFEM domains or general SBFEM domains. Necessary modifications to include forces along the scaling direction, such as body loads, are addressed. The results of the proposed approach are compared to those of traditional FEM models obtained using commercially available software. © 2017 Elsevier Ltdview abstract 10.1016/j.compstruc.2017.06.015 **Coupled acoustic response of two-dimensional bounded and unbounded domains using doubly-asymptotic open boundaries**

Birk, C. and Liu, L. and Song, C.*Journal of Computational Physics*310 (2016)A high-order doubly-asymptotic open boundary for modelling scalar wave propagation in two-dimensional unbounded media is presented. The proposed method is capable of handling domains with arbitrary geometry by using a circular boundary to divide these into near field and far field. The original doubly-asymptotic continued-fraction approach for the far field is improved by introducing additional factor coefficients. Additionally, low-order modes are approximated by singly-asymptotic expansions only to increase the robustness of the formulation. The scaled boundary finite element method is employed to model wave propagation in the near field. Here, the frequency-dependent impedance of bounded subdomains is also expanded into a series of continued fractions. Only three to four terms per wavelength are required to obtain accurate results. The continued-fraction solutions for the bounded domain and the proposed high-order doubly-asymptotic open boundary are expressed in the time-domain as coupled ordinary differential equations, which can be solved by standard time-stepping schemes. Numerical examples are presented to demonstrate the accuracy and robustness of the proposed method, as well as its advantage over existing singly-asymptotic open boundaries. © 2015 Elsevier Inc.view abstract 10.1016/j.jcp.2015.12.029 **Computation of three-dimensional fracture parameters at interface cracks and notches by the scaled boundary finite element method**

Saputra, A.A. and Birk, C. and Song, C.*Engineering Fracture Mechanics*148 (2015)This paper presents the computations of fracture parameters including stress intensity factors and T-stress of three-dimensional cracks and notches by the scaled boundary finite element method. The singular stress field along the crack front is approximated by a singularity at a point through a semi-analytical solution. The solution is expressed as a matrix power function which allows direct extraction of the fracture parameters based on their definitions. No singular element or asymptotic solution is required for the extraction process. The numerical examples presented which include bimaterial interface cracks and V-notches illustrate the accuracy and versatility of the proposed approach. © 2015 Elsevier Ltd.view abstract 10.1016/j.engfracmech.2015.09.006 **Isogeometric analysis enhanced by the scaled boundary finite element method**

Natarajan, S. and Wang, J. and Song, C. and Birk, C.*Computer Methods in Applied Mechanics and Engineering*283 (2015)By leveraging the information of a typical CAD model in the analysis, the intensive process of discretization can be circumvented. This unification has led to the 'Isogeometric Analysis' (IGA) (Hughes etal., 2005). However, as the CAD model provides information only of the boundary, a 2D/3D stress analysis is still one major step away. In this work, the concepts of isogeometric analysis and the scaled boundary finite element method (SBFEM) are combined. The SBFEM requires only the boundary information and hence provides a seamless integration with the CAD modeling. Within the proposed framework, the NURBS basis functions are used to discretize the unknown fields in the circumferential direction, whilst analytical solution is sought in the radial direction. We further extend the framework to problems with singularities and to dynamic analysis. The accuracy and the convergence properties of the proposed method are demonstrated with benchmark problems in the context of linear elasticity and linear elastic fracture mechanics. © 2014 Elsevier B.V.view abstract 10.1016/j.cma.2014.09.003 **Modeling ultrasonic waves in elastic waveguides of arbitrary cross-section embedded in infinite solid medium**

Gravenkamp, H. and Birk, C. and Van, J.*Computers and Structures*149 (2015)An approach is presented to model elastic waveguides of arbitrary cross-section coupled to infinite solid media. The formulation is based on the scaled boundary-finite element method. The surrounding medium is approximately accounted for by a dashpot boundary condition derived from the acoustic impedances of the infinite medium. It is discussed under which circumstances this approximation leads to sufficiently accurate results. Computational costs are very low, since the surrounding medium does not require discretization and the number of degrees of freedom on the cross-section is significantly reduced by utilizing higher-order spectral elements. © 2014 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.compstruc.2014.11.007 **Simulation of elastic guided waves interacting with defects in arbitrarily long structures using the scaled boundary finite element method**

Gravenkamp, H. and Birk, C. and Song, C.*Journal of Computational Physics*295 (2015)In this paper, an approach is presented to model the propagation of elastic waves and their interaction with defects in plate structures. The formulation is based on the Scaled Boundary Finite Element Method (SBFEM), a general semi-analytical method requiring the discretization of boundaries only. For a homogeneous finite or infinite plate section, only the through-thickness direction of the plate is discretized. To describe a defect, the full boundary of a short plate section of irregular shape is discretized. High-order spectral elements are employed for the discretization. The formulation for infinite plates can model the transmission into an unbounded domain exactly. Results are compared with conventional Finite Element Analyses in both time domain and frequency domain. The presented approach allows for the simulation of complex reflection and scattering phenomena using a very small number of degrees of freedom while the mesh consists of one-dimensional elements only. © 2015 Elsevier Inc.view abstract 10.1016/j.jcp.2015.04.032 **Time-domain analysis of wave propagation in 3-D unbounded domains by the scaled boundary finite element method**

Chen, X. and Birk, C. and Song, C.*Soil Dynamics and Earthquake Engineering*75 (2015)Transient wave propagation in three-dimensional unbounded domains is studied. An efficient numerical approach is proposed, which is based on using the displacement unit-impulse response matrix representing the interaction force-displacement relationship on the near field/far field interface. Spatially, an approximation is used to reduce the computational effort associated with the large size of three-dimensional problems. It is based on subdividing the fully coupled unbounded domain into multiple subdomains. The displacement unit-impulse response matrices of all subdomains are calculated separately. The error associated with this spatial decoupling can be reduced by placing the near field/far field interface further away from the domain of interest. Detailed parameter studies have been conducted using numerical examples, in order to provide guidelines for the proposed spatially local schemes, and to demonstrate the accuracy and high efficiency of the proposed method for three-dimensional soil-structure interaction problems. © 2015 Elsevier Ltd.view abstract 10.1016/j.soildyn.2015.04.009 **Transient analysis of wave propagation in layered soil by using the scaled boundary finite element method**

Chen, X. J. and Birk, C. and Song, C. M.*Computers and Geotechnics*63 (2015)An efficient method for modelling the propagation of elastic waves in layered media is developed. It is applicable to scalar and vector wave soil-structure interaction problems involving semi-infinite layers. The scaled boundary finite element method is employed to derive an equation for the displacement unit-impulse response matrix on the near field/far field interface. An accurate and efficient time discretization method is proposed for that equation. As the displacement unit-impulse response approaches zero, the convolution integral representing the force-displacement relationship can be truncated. After the truncation the computational effort only increases linearly with time. Thus, a considerable reduction of computational effort is achieved in a time domain analysis. In addition, a reasonable viscous damping model is proposed for this problem. The existence of damping will cause the displacement unit-impulse response matrix to decay faster. Therefore, an earlier truncation time can be adopted, thus further reducing the computational effort. Numerical examples demonstrate the accuracy and high efficiency of the new method. (C) 2014 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.compgeo.2014.08.008 **A high-order approach for modelling transient wave propagation problems using the scaled boundary finite element method**

Chen, D. and Birk, C. and Song, C. and Du, C.*International Journal for Numerical Methods in Engineering*97 (2014)A high-order time-domain approach for wave propagation in bounded and unbounded domains is proposed. It is based on the scaled boundary FEM, which excels in modelling unbounded domains and singularities. The dynamic stiffness matrices of bounded and unbounded domains are expressed as continued-fraction expansions, which leads to accurate results with only about three terms per wavelength. An improved continued-fraction approach for bounded domains is proposed, which yields numerically more robust time-domain formulations. The coefficient matrices of the corresponding continued-fraction expansion are determined recursively. The resulting solution is suitable for systems with many DOFs as it converges over the whole frequency range, even for high orders of expansion. A scheme for coupling the proposed improved high-order time-domain formulation for bounded domains with a high-order transmitting boundary suggested previously is also proposed. In the time-domain, the coupled model corresponds to equations of motion with symmetric, banded and frequency-independent coefficient matrices, which can be solved efficiently using standard time-integration schemes. Numerical examples for modal and time-domain analysis are presented to demonstrate the increased robustness, efficiency and accuracy of the proposed method. © 2013 John Wiley & Sons, Ltd.view abstract 10.1002/nme.4613 **A high-order doubly asymptotic open boundary for scalar waves in semi-infinite layered systems**

Prempramote, S. and Birk, C. and Song, Ch.*IOP Conference Series: Materials Science and Engineering*10 (2014)Wave propagation in semi-infinite layered systems is of interest in earthquake engineering, acoustics, electromagnetism, etc. The numerical modelling of this problem is particularly challenging as evanescent waves exist below the cut-off frequency. Most of the high-order transmitting boundaries are unable to model the evanescent waves. As a result, spurious reflection occurs at late time. In this paper, a high-order doubly asymptotic open boundary is developed for scalar waves propagating in semi-infinite layered systems. It is derived from the equation of dynamic stiffness matrix obtained in the scaled boundary finite-element method in the frequency domain. A continued-fraction solution of the dynamic stiffness matrix is determined recursively by satisfying the scaled boundary finite-element equation at both high- and low-frequency limits. In the time domain, the continued-fraction solution permits the force-displacement relationship to be formulated as a system of first-order ordinary differential equations. Standard time-step schemes in structural dynamics can be directly applied to evaluate the response history. Examples of a semi-infinite homogeneous layer and a semi-infinite two-layered system are investigated herein. The displacement results obtained from the open boundary converge rapidly as the order of continued fractions increases. Accurate results are obtained at early time and late time. © 2010 IOP Publishing Ltd.view abstract 10.1088/1757-899X/10/1/012215 **A physically and geometrically nonlinear scaled-boundary-based finite element formulation for fracture in elastomers**

Behnke, R. and Mundil, M. and Birk, C. and Kaliske, M.*International Journal for Numerical Methods in Engineering*99 (2014)This paper is devoted to the formulation of a plane scaled boundary finite element with initially constant thickness for physically and geometrically nonlinear material behavior. Special two-dimensional element shape functions are derived by using the analytical displacement solution of the standard scaled boundary finite element method, which is originally based on linear material behavior and small strains. These 2D shape functions can be constructed for an arbitrary number of element nodes and allow to capture singularities (e.g.,at a plane crack tip) analytically, without extensive mesh refinement. Mapping these proposed 2D shape functions to the 3D case, a formulation that is compatible with standard finite elements is obtained. The resulting physically and geometrically nonlinear scaled boundary finite element formulation is implemented into the framework of the finite element method for bounded plane domains with and without geometrical singularities. The numerical realization is shown in detail. To represent the physically and geometrically nonlinear material and structural behavior of elastomer specimens, the extended tube model and the Yeoh model are used. Numerical studies on the convergence behavior and comparisons with standard Q1P0 finite elements demonstrate the correct implementation and the advantages of the developed scaled boundary finite element. © 2014 John Wiley & Sons, Ltd.view abstract 10.1002/nme.4714 **Computation of dispersion curves for embedded waveguides using a dashpot boundary condition**

Gravenkamp, H. and Birk, C. and Song, C.*Journal of the Acoustical Society of America*135 (2014)In this paper a numerical approach is presented to compute dispersion curves for solid waveguides coupled to an infinite medium. The derivation is based on the scaled boundary finite element method that has been developed previously for waveguides with stress-free surfaces. The effect of the surrounding medium is accounted for by introducing a dashpot boundary condition at the interface between the waveguide and the adjoining medium. The damping coefficients are derived from the acoustic impedances of the surrounding medium. Results are validated using an improved implementation of an absorbing region. Since no discretization of the surrounding medium is required for the dashpot approach, the required number of degrees of freedom is typically 10 to 50 times smaller compared to the absorbing region. When compared to other finite element based results presented in the literature, the number of degrees of freedom can be reduced by as much as a factor of 4000. © 2014 Acoustical Society of America.view abstract 10.1121/1.4864303 **Dynamic response of foundations on three-dimensional layered soil using the scaled boundary finite element method**

Birk, C. and Behnke, R.*IOP Conference Series: Materials Science and Engineering*10 (2014)This paper is devoted to the dynamic analysis of arbitrarily shaped three-dimensional foundations on layered ground using a coupled FEM-SBFEM approach. A novel scaled boundary finite element method for the analysis of three-dimensional layered continua over rigid bedrock is derived. The accuracy of the new method is demonstrated using rigid circular foundations resting on or embedded in nonhomogeneous soil layers as examples. © 2010 IOP Publishing Ltd.view abstract 10.1088/1757-899X/10/1/012228 **Numerical modeling of elastic waveguides coupled to infinite fluid media using exact boundary conditions**

Gravenkamp, H. and Birk, C. and Song, C.*Computers and Structures*141 (2014)The simulation of guided waves in plate structures and cylinders coupled to infinite fluids is addressed. The approach is based on the Scaled Boundary Finite Element Method. Only a straight line is discretized that represents the through-thickness direction or the radial direction. The surrounding fluid is accounted for by employing a damping boundary condition that is based on the analytical description of the radiation impedance. Since the radiation impedance is a function of the wavenumber in the waveguide, an iterative solution procedure is applied. The algorithm is highly efficient while the results are in agreement with the Global Matrix Method. © 2014 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.compstruc.2014.05.010 **Numerical modelling of wave propagation in anisotropic soil using a displacement unit-impulse-response-based formulation of the scaled boundary finite element method**

Chen, X. and Birk, C. and Song, C.*Soil Dynamics and Earthquake Engineering*65 (2014)An efficient method for modelling the propagation of elastic waves in unbounded domains is developed. It is applicable to soil-structure interaction problems involving scalar and vector waves, unbounded domains of arbitrary geometry and anisotropic soil. The scaled boundary finite element method is employed to derive a novel equation for the displacement unit-impulse response matrix on the soil-structure interface. The proposed method is based on a piecewise linear approximation of the first derivative of the displacement unit-impulse response matrix and on the introduction of an extrapolation parameter in order to improve the numerical stability. In combination, these two ideas allow for the choice of significantly larger time steps compared to conventional methods, and thus lead to increased efficiency. As the displacement unit-impulse response approaches zero, the convolution integral representing the force-displacement relationship can be truncated. After the truncation the computational effort only increases linearly with time. Thus, a considerable reduction of computational effort is achieved in a time domain analysis. Numerical examples demonstrate the accuracy and high efficiency of the new method for two-dimensional soil-structure interaction problems. © 2014 Elsevier Ltd.view abstract 10.1016/j.soildyn.2014.06.019 **The computation of dispersion relations for axisymmetric waveguides using the Scaled Boundary Finite Element Method**

Gravenkamp, H. and Birk, C. and Song, C.*Ultrasonics*54 (2014)This paper addresses the computation of dispersion curves and mode shapes of elastic guided waves in axisymmetric waveguides. The approach is based on a Scaled Boundary Finite Element formulation, that has previously been presented for plate structures and general three-dimensional waveguides with complex cross-section. The formulation leads to a Hamiltonian eigenvalue problem for the computation of wavenumbers and displacement amplitudes, that can be solved very efficiently. In the axisymmetric representation, only the radial direction in a cylindrical coordinate system has to be discretized, while the circumferential direction as well as the direction of propagation are described analytically. It is demonstrated, how the computational costs can drastically be reduced by employing spectral elements of extremely high order. Additionally, an alternative formulation is presented, that leads to real coefficient matrices. It is discussed, how these two approaches affect the computational efficiency, depending on the elasticity matrix. In the case of solid cylinders, the singularity of the governing equations that occurs in the center of the cross-section is avoided by changing the quadrature scheme. Numerical examples show the applicability of the approach to homogeneous as well as layered structures with isotropic or anisotropic material behavior. © 2014 Elsevier B.V. All rights reserved.view abstract 10.1016/j.ultras.2014.02.004 **A modified scaled boundary finite element method for three-dimensional dynamic soil-structure interaction in layered soil**

Birk, C. and Behnke, R.*International Journal for Numerical Methods in Engineering*89 (2012)This paper is devoted to the analysis of elastodynamic problems in 3D-layered systems which are unbounded in the horizontal direction. For this purpose, a finite element model of the near field is coupled to a scaled boundary finite element model (SBFEM) of the far field. The SBFEM is originally based on describing the geometry of a half-space or full-space domain by scaling the geometry of the near field/far field interface using a radial coordinate. A modified form of the SBFEM for waves in a 2D layer is also available. None of these existing formulations can be used to describe a 3D-layered medium. In this paper, a modified SBFEM for the analysis of 3D-layered continua is derived. Based on the use of a scaling line instead of a scaling centre, a suitable scaled boundary transformation is proposed. The derivation of the corresponding scaled boundary finite element (SBFE) equations in displacement and stiffness is presented in detail. The latter is a nonlinear differential equation with respect to the radial coordinate, which has to be solved numerically for each excitation frequency considered in the analysis. Various numerical examples demonstrate the accuracy of the new method and its correct implementation. These include rigid circular and square foundations embedded in or resting on the surface of layered homogeneous or inhomogeneous 3D soil deposits over rigid bedrock. Hysteretic damping is assumed in some cases. The dynamic stiffness coefficients calculated using the proposed method are compared with analytical solutions or existing highly accurate numerical results. © 2011 John Wiley & Sons, Ltd.view abstract 10.1002/nme.3251 **An improved continued-fraction-based high-order transmitting boundary for time-domain analyses in unbounded domains**

Birk, C. and Prempramote, S. and Song, C.*International Journal for Numerical Methods in Engineering*89 (2012)A high-order local transmitting boundary to model the propagation of acoustic or elastic, scalar or vector-valued waves in unbounded domains of arbitrary geometry is proposed. It is based on an improved continued-fraction solution of the dynamic stiffness matrix of an unbounded medium. The coefficient matrices of the continued-fraction expansion are determined recursively from the scaled boundary finite element equation in dynamic stiffness. They are normalised using a matrix-valued scaling factor, which is chosen such that the robustness of the numerical procedure is improved. The resulting continued-fraction solution is suitable for systems with many DOFs. It converges over the whole frequency range with increasing order of expansion and leads to numerically more robust formulations in the frequency domain and time domain for arbitrarily high orders of approximation and large-scale systems. Introducing auxiliary variables, the continued-fraction solution is expressed as a system of linear equations in iω in the frequency domain. In the time domain, this corresponds to an equation of motion with symmetric, banded and frequency-independent coefficient matrices. It can be coupled seamlessly with finite elements. Standard procedures in structural dynamics are directly applicable in the frequency and time domains. Analytical and numerical examples demonstrate the superiority of the proposed method to an existing approach and its suitability for time-domain simulations of large-scale systems. © 2011 John Wiley & Sons, Ltd.view abstract 10.1002/nme.3238 **A local high-order doubly asymptotic open boundary for diffusion in a semi-infinite layer**

Birk, C. and Song, C.*Journal of Computational Physics*229 (2010)A high-order open boundary for transient diffusion in a semi-infinite homogeneous layer is developed. The method of separation of variables is used to derive a relationship between the modal function and the flux at the near field/far field boundary in the Fourier domain. The resulting equation in terms of the modal impedance coefficient is solved by expanding the latter into a doubly asymptotic series of continued fractions. As a result, the open boundary condition in the Fourier domain is represented by a system of algebraic equations in terms of . iω. This corresponds to a system of fractional differential equations of degree . α=. 0.5 in the time-domain. This temporally global formulation is transformed into a local description by introducing internal variables. The resulting local high-order open boundary condition is highly accurate, as is demonstrated by a number of heat transfer examples. A significant gain in accuracy is obtained in comparison with existing singly-asymptotic formulations at no additional computational cost. © 2010 Elsevier Inc.view abstract 10.1016/j.jcp.2010.04.046 **A temporally local absorbing boundary for diffusion in 3D unbounded domains**

Birk, C. and Song, C.*AIP Conference Proceedings*1233 (2010)The scaled boundary finite element method is used to model diffusion in unbounded domains. A time-domain representation is derived expanding the stiffness into a series of continued fractions and using fractional derivatives. A method for transforming the resulting system of fractional differential equations into a local formulation is presented. © 2010 American Institute of Physics.view abstract 10.1063/1.3452234 **An improved non-classical method for the solution of fractional differential equations**

Birk, C. and Song, C.*Computational Mechanics*46 (2010)A procedure to construct temporally local schemes for the computation of fractional derivatives is proposed. The frequency-domain counterpart (iω) α of the fractional differential operator of order α is expressed as an improper integral of a rational function in iω. After applying a quadrature rule, the improper integral is approximated by a series of partial fractions. Each term of the partial fractions corresponds to an exponential kernel in the time domain. The convolution integral in a fractional derivative can be evaluated recursively leading to a local scheme. As the arguments of the exponential functions are always real and negative, the scheme is stable. The present procedure provides a convenient way to evaluate the quality of a given algorithm by examining its accuracy in fitting the function (iω) α. It is revealed that the non-classical solution methods for fractional differential equations proposed by Yuan and Agrawal (ASME J Vib Acoust 124:321-324, 2002) and by Diethelm (Numer Algorithms 47:361-390, 2008) can also be interpreted as applying specific quadrature rules to evaluate the improper integral numerically. Over a wider range of frequencies, Diethelm's algorithm provides a more accurate fitting than the YA algorithm. Therefore, it leads to better performance. Further exploiting this advantage of the proposed derivation, a novel quadrature rule leading to an even better performance than Diethelm's algorithm is proposed. Significant gains in accuracy are achieved at the extreme ends of the frequency range. This results in significant improvements in accuracy for late time responses. Several numerical examples, including fractional differential equations of degree α = 0.3 and α = 1.5, demonstrate the accuracy and efficiency of the proposed method. © Springer-Verlag 2010.view abstract 10.1007/s00466-010-0510-4 **Mmtf: The Maryland-magellan Tunable Filter**

Veilleux, S. and Weiner, B. J. and Rupke, D. S. N. and McDonald, M. and Birk, C. and Bland-Hawthorn, J. and Dressler, A. and Hare, T. and Osip, D. and Pietraszewski, C. and Vogel, S. N.*Astronomical Journal*139 (2010)This paper describes the Maryland-Magellan Tunable Filter (MMTF) on the Magellan-Baade 6.5 m telescope. MMTF is based on a 150 mm clear aperture Fabry-Perot (FP) etalon that operates in low orders and provides transmission bandpass and central wavelength adjustable from similar to 5 angstrom to similar to 15 angstrom and from similar to 5000 angstrom to over similar to 9200 angstrom, respectively. It is installed in the Inamori Magellan Areal Camera and Spectrograph and delivers an image quality of similar to 0.'' 5 over a field of view of 27' in diameter (monochromatic over similar to 10'). This versatile and easy-to-operate instrument has been used over the past three years for a wide variety of projects. This paper first reviews the basic principles of FP tunable filters, and then provides a detailed description of the hardware and software associated with MMTF and the techniques developed to observe with this instrument and reduce the data. The main lessons learned in the course of the commissioning and implementation of MMTF are highlighted next, before concluding with a brief outlook on the future of MMTF and of similar facilities which are soon coming on line.view abstract 10.1088/0004-6256/139/1/145

#### computational mechanics

#### elastostatics and elastodynamics

#### finite element method

#### mesh based methods

#### numerical methods

#### structural analysis