The scaled boundary finite element method (SBFEM) excels as a tool for numerical analysis at particular problem setups where the analytical solution in the scaling direction can be exploited to improve computational efficiency by reducing the number of required degrees of freedom (DOF). This is especially the case for simulating axisymmetric waveguides in the high-frequency range, allowing a significant decrease of computational costs (both memory and CPU time). Then, only the radial direction in a cylindrical coordinate system is discretized and the axial direction is solved analytically. A full threedimensional formulation is possible via the Fourier transform to include asymmetries. This contribution presents such an axisymmetric formulation, which is extended to allow the definition of circumferential as well as arbitrarily shaped dynamic boundary conditions (BCs). Furthermore, the required number of DOF depends on the frequency content. Hierarchical shape functions allow to dynamically adapt the DOF, further increasing efficiency. It will be shown that the results are in good agreement with standard finite element procedures, while greatly reducing computational time

Dreiling, Dmitrij

Feldmann, Nadine

Gravenkamp, Hauke

Henning, Bernd

Itner, Dominik

Scipedia

14th World Congress on Computational Mechanics (WCCM)ECCOMAS Congress 2020)Virtual Congress: 11-–15 January 2021F. Chinesta, R. Abgrall, O. Allix and M. Kaliske (Eds)SIMULATION OF GUIDED WAVES IN CYLINDERS SUBJECT TOARBITRARY BOUNDARY CONDITIONS USING THE SCALEDBOUNDARY FINITE ELEMENT METHODDominik Itner1, Hauke Gravenkamp1, Dmitrij Dreiling2, Nadine Feldmann2 AND BerndHenning21 Department of Civil Engineering, University of Duisburg-Essen45141 Essen, Germany{dominik.itner|hauke.gravenkamp}@uni-due.de2 Measurement Engineering Group, Paderborn University33098 Paderborn, Germany{dreiling|feldmann|henning}@emt.uni-paderborn.deKey words: ultrasound, guided waves, wave propagation, hierarchical shape functions, scaled boundaryfinite element methodAbstract. The scaled boundary finite element method (SBFEM) excels as a tool for numerical analysisat particular problem setups where the analytical solution in the scaling direction can be exploited toimprove computational efficiency by reducing the number of required degrees of freedom (DOF). Thisis especially the case for simulating axisymmetric waveguides in the high-frequency range, allowing asignificant decrease of computational costs (both memory and CPU time). Then, only the radial directionin a cylindrical coordinate system is discretized and the axial direction is solved analytically. A full three-dimensional formulation is possible via the Fourier transform to include asymmetries. This contributionpresents such an axisymmetric formulation, which is extended to allow the definition of circumferentialas well as arbitrarily shaped dynamic boundary conditions (BCs). Furthermore, the required number ofDOF depends on the frequency content. Hierarchical shape functions allow to dynamically adapt theDOF, further increasing efficiency. It will be shown that the results are in good agreement with standardfinite element procedures, while greatly reducing computational time.1 INTRODUCTIONUltrasound simulations of soft materials such as polymers usually require a large number of DOF toproperly capture the displacement solution when using standard FEM applications. This comes at thedetriment of memory usage and computation times and is especially difficult to handle given long struc-tures or unbounded domains. For such applications, the SBFEM has been proven a useful tool solvinglinear elastic problems in the frequency domain [1, 3].When considering an axisymmetric domain along a cylindrical coordinate system, only the radial direc-tion is discretized whereas an analytical solution is considered in the axial direction. Such a model can beextended in the SBFEM to include asymmetries in BCs allowing a full three-dimensional description ofthe problem by using concepts of the Fourier FEM. Then, the system is approximated w.r.t. the circum-1Dominik Itner, Hauke Gravenkamp, Dmitrij Dreiling, Nadine Feldmann and Bernd Henningference by a truncated Fourier series and solved for each spatial frequency. Likewise, such a dynamicasymmetric BC must then be transformed, both in time and space. An arbitrarily shaped BC can varywith the radius and circumference and as such must be sampled at several distinct radii and transformedindividually while still retaining computational efficiency.In this paper, we briefly introduce the SBFEM for cylindric domains leading to the dynamic stiffnessmatrix. Subsequently, we present the preprocessing for applying an arbitrarily spatially varying BC;especially in the context of hierarchical shape functions. We verify the proposed approach by comparisonwith a finite element-model and demonstrate that a significant speed-up can be achieved for dynamicsimulations.2 THE SBFEM FORMULATIONA short summary of the cylindric SBFEM formulation is presented here. The displacement-strain rela-tionship is decomposed into its constituent differentials and processed individuallyε= Lu = b1∂zu+1rb2∂θ u+b3∂ru+1rb4u = B1∂zû+B2û , (1)which leads to matrices B1 and B2 given polar number m (circumferential frequency), additionally usinga discretization with the matrix of shape functions N for the discrete displacement vector û [2]:B1 = b1N and B2 =1r(imb2 +b4)N+2hb3∂ηN , m ∈N0 . (2)Applying the principle of virtual work leads to the coefficient matrices E0, E1, E2, and M0E0 =∫ +1−1BH1 DB1|J|dη , E1 =∫ +1−1BH2 DB1|J|dη , E2 =∫ +1−1BH2 DB2|J|dη , (3a)M0 = ρ∫ +1−1NHN|J|dη (3b)in the following partial differential equation:E0∂zzû+(EH1 −E1)∂zû−E2û−M0∂tt û = 0 ⇒ [λ 2E0 +λ (EH1 −E1)−E2−ω2M0] û = 0 , (4)which can be solved in the frequency domain with angular frequencies ω for eigenvalues λ . Likewise,the coefficient matrix Z withZ(m,ω) =[E−10 EH1 −E−10ω2M0 +E1E−10 EH1 −E2 −E1E−10](5)can be derived, leading to the ordinary matrix differential equation and its general solution [2]∂zφ(z) =−Zφ(z) ⇒ φ(z) = ΨeΛzc with φ(z) =[û(z)q̂(z)](6)to the eigenvalue problem −ZΨ = ΨΛ. Here, Ψ is the eigenvector matrix associated with the diagonaleigenvalue matrix Λ, and c is the vector of integration constants. Partitioning the solution in Eq. (6)2Dominik Itner, Hauke Gravenkamp, Dmitrij Dreiling, Nadine Feldmann and Bernd Henningleads to the dynamic stiffness matrix of a bounded domain of arbitrary length L [3] of a solid or hollowcylinderSL(m,ω) =[−Ψqp −Ψqne−ΛnLΨqpeΛpL Ψqn][Ψup Ψune−ΛnLΨupeΛpL Ψun]−1, (7)which is given for each combination of ω and m and defines the relationship between displacements andexternal forces at the opposing cylinder faces:[p̂(m,ω)|z=0p̂(m,ω)|z=L]= SL(m,ω)[û(m,ω)|z=0û(m,ω)|z=L]. (8)Boundary conditions may then be prescribed along the discretized flat surfaces either at z = 0 or z = Land the linear equation system in Eq. (8) solved as per usual.3 ARBITRARILY SHAPED BOUNDARY CONDITIONGiven that the stiffness matrix and consequently the system of linear equations is solved in the frequencydomain, any dynamic BC with an asymmetric circumferential distribution must be similarly Fouriertransformed F(m,ω) = F ( f (θ , t)). In the trivial case, the distribution of a BC may be assumed to havefollowing property:f (r,θ , t) = f (r) · f (θ) · f (t) , (9)which allows to individually perform a one-dimensional Fourier transform of f (θ) and f (t), respec-tively,1 as well as integration along the radius in the case of distributed loads. In contrast, arbitrarilyshaped BCs lose the independency of r and θ givenf (r,θ , t) = f (r,θ) · f (t) , (10)introducing a relationship between the radial and circumferential distribution, i.e. an arbitrarily shapeddistribution on the flat cylinder surface. It then becomes necessary to sample such a distribution alongmultiple concentric circles with varying radii rj to correctly discretize and transform it:F(rj,m) =12π∫ 2π0f (rj,θ) · e−imθ dθ , m ∈N0 . (11)Each concentric distribution is independently Fourier-transformed to finally assemble the vector of dis-crete values for a particular polar number m (and angular frequency ω):f̂ =[f̂1 . . . f̂j . . . f̂n]T with f̂j = F(rj,m) ·F(ω) . (12)The radii rj are chosen to coincide with the position of nodes if the discretization is based on shape func-tions that possess the Kronecker delta property. It may also be necessary to select the integration points,instead, depending on the integration scheme for loads. However, using hierarchical shape functions, amore involved process is required, as the discrete values are not related to physical positions but ratherrepresent the amplitude spectrum of the polynomial basis functions and hence require conversion. Then,1Due to symmetry reasons, the number of coefficients for which the system must be solved is halved, not quartered.3Dominik Itner, Hauke Gravenkamp, Dmitrij Dreiling, Nadine Feldmann and Bernd Henning(a) (b)Figure 1: The displacement solution to the static case for a Neumann boundary condition demonstratesthe error comparing an insufficient equidistant sampling (a) and ”well-behaved” sampling using Gauss-Lobatto-Legendre points (b).the choice of rj is not trivial and can introduce errors in the approximation and therefore the solution.These errors are expressed as edge effects which originate in Runge’s phenomenon (as circumvented byspectral elements) and especially occur using equidistant positions [2, 4]; additionally, these amplifica-tions of interpolated values are smeared along the circumference due to the Fourier transform, cf. Fig. 1aand Fig. 1b. As such, an adequate choice of radii rj is required.It was found that when Lobatto polynomials (up to order p) are used as the near-orthogonal basis func-tions, withNk(η) =12(1−η) , k = 012(1+η) , k = 1√2k−12∫η−1Lk−1(ζ )dζ , k ∈ [2, p](13)(here, Lk−1 refers to the Legendre polynomials), then likewise Gauss-Lobatto-Legendre points providea good choice to circumvent Runge’s phenomenon and minimize the approximation error. A mappingbetween the physical domain and the polynomial coefficient space can be constructed by evaluating theLobatto polynomials at local coordinates η j that correspond to rj with n = p+1N̂ =N1(η1) . . . Nk(η1) . . . Nn(η1).........N1(η j) . . . Nk(η j) . . . Nn(η j).........N1(ηn) . . . Nk(ηn) . . . Nn(ηn) , (14)and subsequently inverting the matrix N̂:f̂(m) = N̂−1F(r,m) , rT =[r1 . . . rj . . . rn]. (15)4Dominik Itner, Hauke Gravenkamp, Dmitrij Dreiling, Nadine Feldmann and Bernd HenningFigure 2: Geometry of hollow cylinder. Load isapplied on ellipse area. Displacements uz are fixedat the base. Sample point of interest is indicated inpoint A with r = 1.05mm and θ = 1°.Figure 3: Dynamic response uz of FEM () andSBFEM (#) in point A shown in Fig. 2.The matrix N̂ is not strictly required to be a square matrix, however, undersampling is not recommended.In case of oversampling, the pseudoinverse must necessarily be applied. When external forces are pre-scribed, additional pre-integration of shape functions is necessary to assemble a Neumann BC:p̂ =(∫ +1−1NHN|J|dη)f̂ . (16)This concludes the application of arbitrarily shaped BCs.4 NUMERICAL EXAMPLEThe following example is chosen to demonstrate the application of an arbitrarily shaped BC. The area ofeffect is confined to a rotated ellipse placed on the top surface of a cylinder. A Neumann BC is appliedgiven a parabolic distribution around the center of the ellipse in the local coordinate system (R,ϕ), seeFig. 2. The following Gauss-modulated sine function is chosen as the dynamic signal:f (t) = sin(2π ·1MHz · t) · exp(−12[t−3µs0.5µs]2)· (1GPa) . (17)Additionally, displacements in z are fixed at the bottom. The displacements are evaluated at z = 0.5mmand the L2-norm is applied to compute the overall deviation averaged in space and time. As reference thesolution aquired by Ansys 2019 R2 is chosen using implicit time integration. There, the mesh is definedby an average element size of 0.1mm (roughly 10 elements in r,z and 128 elements in θ ), and a time stepof 0.04 µs with a time window of interest of 6 µs. For the convergence study, each value is halved for atotal of four simulations. The SBFEM model, on the other hand, requires only a minimum and maximumnumber of 15 and 17 DOF per discretized edge, respectively, to reach a precision comparable to thereference. Here, we apply an adaptive algorithm to automatically pick the number of DOF dependingon the current frequencies ω and m, maintaining a predefined error. However, the minimum number isincreased to properly capture the BC.Selecting 5289 unique Fourier coefficients to compute the solution, a computing time of 89.4s is re-quired and yields a speedup of over 2000 compared to the third Ansys solution (mesh size: 0.025 mm,5Dominik Itner, Hauke Gravenkamp, Dmitrij Dreiling, Nadine Feldmann and Bernd Henning(a) (b)Figure 4: Convergence behavior of FEM (a) using h-refinement and SBFEM (b) using p-refinement for2952 (), 4428 (), and 5289 ( ) Fourier coefficients. Computation times are indicated. Both solutionsuse a fourth FEM simulation as reference (with roughly seven weeks computation time).time step: 0.01 µs). Relative differences of 0.8684%, 0.9107%, and 0.8369% in r, θ , and z, respectively,can be found between the methods. Figure 3 shows the dynamic response in uz at the single point indi-cated in Fig. 2, which possesses the largest relative deviation of all sample points with 0.8090 % for thesimulated time frame. As can be seen, the solutions are in very good agreement.A more thorough comparison is difficult because errors are not only introduced due to discretizationbut also due to truncation of the Fourier series. This interplay of approximation errors becomes evidentwhen the convergence for different sets of Fourier coefficients is plotted. For this case, we opt to adaptthe previously mentioned algorithm such that fewer than the minimum required DOF (due to the BC)is allowed. (This explains differences in computation times, even though results still match very well.)As can be seen, proper convergence behavior strongly depends on the number of Fourier coefficients forwhich the system is solved, see Fig. 4. This analysis primarily regards model behavior in the physicaldomain (including both time and space) and shows that the model requires not only a fine enough dis-cretization but also enough information in the frequency domain to capture the solution well enough inthe physical domain.5 CONCLUSIONIn this contribution, the cylindric SBFEM was presented and arbitrarily shaped BC were incorporated. Itwas demonstrated that the formulation gives good results while requiring only a fraction of the computingtime of conventional FEM applications.Acknowledgement Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -409779252.6Dominik Itner, Hauke Gravenkamp, Dmitrij Dreiling, Nadine Feldmann and Bernd HenningREFERENCES[1] Song, C. and Wolf, J. P. The scaled boundary finite-element method—alias consistent infinitesimalfinite-element cell method—for elastodynamics. Computer Methods in Applied Mechanics and En-gineering (1997) 147(3):329–355.[2] Gravenkamp, H. and Birk, C. and Song, C. The computation of dispersion relations for axisymmet-ric waveguides using the Scaled Boundary Finite Element Method. Ultrasonics (2014) 54(5):1373–1385.[3] Gravenkamp, H. and Birk, C. and Song, C. Simulation of elastic guided waves interacting withdefects in arbitrarily long structures using the Scaled Boundary Finite Element Method. Journal ofComputational Physics (2015) 295:438–455.[4] Vu, T.H. and Deeks, A.J. Use of higher-order shape functions in the scaled boundary finite elementmethod. Int. J. Numer. Methods Eng. (2006) 65(10):1714–1733.7

Simulation of Guided Waves in Cylinders Subject to Arbitrary Boundary Conditions Using the Scaled Boundary Finite Element Method

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Simulation of Guided Waves in Cylinders Subject to Arbitrary Boundary Conditions Using the Scaled Boundary Finite Element Method

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