Abstract
In this chapter a recently developed novel approach to simulate the propagation of ultrasonic guided waves in heterogeneous, especially cellular lightweight structures is presented. One of the most important drawbacks of traditional finite element-based approaches is the need for a geometry-conforming discretization. It is generally acknowledged that the mesh generation process constitutes the bottleneck in the current simulation pipeline. Therefore, different measures have been taken to at least alleviate the meshing bruden. One of these attempts is the finite cell method (FCM). It combines the advantages known from higher order finite element methods (FEM; exponential convergence rates) with those of fictitious domain methods (FDM; automatic mesh generation using Cartesian grids). In the framework of the FCM we do not rely on body-fitted discretizations and therefore shift the effort typically required for the mesh generation to the numerical integration of the system matrices which is performed by means of an adaptive Gaussian quadrature. The advantage of such a procedure in the context of wave propagation analysis is seen in the fully automated analysis process. Consequently, hardly any user input is required making the simulation very robust. Notwithstanding, it can be mathematically proven that the optimal rates of convergence are retained.
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References
Abedian A, Parvizian J, Düster A, Rank E (2013) The finite cell method for the J 2 flow theory of plasticity. Finite Elem Anal Des 69:37–47
Abedian A, Parvizian J, Düster A, Rank E (2014) The FCM compared to the h-version FEM for elasto-plastic problems. Appl Math Mech 35:1239–1248
Almeida JPM, Pereira OJBA (1996) A set of hybrid equilibrium finite element models for the analysis of three-dimensional solids. Int J Numer Methods Eng 39:2789–2802
Belytschko T, Chen JS (2009) Meshfree and particle methods. Wiley, Chichester
Blyth MG, Pozrikidis C (2006) A Lobatto interpolation grid over the triangle. IMA J Appl Math 71:153–169
Cattani C, Paoluzzi A (1990) Boundary integration over linear polyhedra. Comput Aided Des 22:130–135
Cohen E, Martin T, Kirby RM, Lyche T, Riesenfeld RF (2010) Analysis-aware modeling: understanding quality considerations in modeling for isogeometric analysis. Comput Methods Appl Mech Eng 199:334–356
Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley, Hoboken
Dasgupta G (2003) Integration within polygonal finite elements. J Aerosp Eng 16:9–18
Dauge M, Düster A, Rank E (2015) Theoretical and numerical investigation of the finite cell method. J Sci Comput 65:1039–1064
Duarte C, Babuška I, Oden J (2000) Generalized finite element method for three-dimensional structural mechanics problems. Comput Struct 77(2):215–232
Duczek S (2014) Higher order finite elements and the fictitious domain concept for wave propagation analysis. VDI Fortschritt-Berichte Reihe 20 Nr. 458
Duczek S, Gabbert U (2015) Efficient integration method for fictitious domain approaches. Comput Mech 56:725–738
Duczek S, Gabbert U (2016) The finite cell method for polygonal meshes: polygonal-FCM. Comput Mech 58:587–618
Duczek S, Joulaian M, Düster A, Gabbert U (2014) Numerical analysis of Lamb waves using the finite and spectral cell methods. Int J Numer Methods Eng 99:26–53
Duczek S, Liefold S, Gabbert U (2015) The finite and spectral cell methods for smart structure applications: transient analysis. Acta Mech 226:845–869
Duczek S, Duvigneau F, Gabbert U (2016) The finite cell method for arbitrary tetrahedral meshes. Finite Elem Anal Des 121:18–32
Dumonet D (2014) Towards efficient and accurate 3d cut cell integration in the context of the finite cell method. Master’s thesis, Technical University Munich
Düster A, Niggl A, Rank E (2007) Applying the hp-d version of the FEM to locally enhance dimensionally reduced models. Comput Methods Appl Mech Eng 196:3524–3533
Düster A, Parvizian J, Yang Z, Rank E (2008) The finite cell method for three-dimensional problems of solid mechanics. Comput Methods Appl Mech Eng 197:3768–3782
Düster A, Sehlhorst HG, Rank E (2012) Numerical homogenization of heterogeneous and cellular materials utilizing the finite cell method. Comput Mech 1:1–19
Gao XW (2002) The radial integration method for evaluation of domain integrals with boundary-only discretization. Eng Anal Bound Elem 26:905–916
Gonzales-Ochoa C, McCammon S, Peters J (1998) Computing moments of objects enclosed by piecewise polynomial surfaces. ACM Trans Graph 17:143–157
Hematiyan MR (2007) A general method for evaluation of 2d and 3d domain integrals without domain discretization and its application in BEM. Comput Mech 39:509–520
Hubrich S, Joulaian M, Düster A (2015) Numerical integration in the finite cell method based on moment-fitting. In: Proceedings of the 3rd ECCOMAS young investigators conference and 6th GACM colloquium – YIC GACM 2015
Joulaian M, Düster A (2013) Local enrichment of the finite cell method for problems with material interfaces. Comput Mech 52:741–762
Joulaian M, Duczek S, Gabbert U, Düster A (2014) Finite and spectral cell method for wave propagation in heterogeneous materials. Comput Mech 54:661–675
Kaufmann P, Martin S, Botsch M, Gross M (2009) Flexible simulation of deformable models using discontinuous Galerkin FEM. Graph Model 71:153–167
Komatitsch D, Tromp J (2002) Spectral-element simulations of global seismic wave propagation I. – Validation. Int J Geophys 149:390–412
Komatitsch D, Tromp J (2002) Spectral-element simulations of global seismic wave propagation II. – Three-dimensional models, oceans, rotation and self-gravitation. Int J Geophys 150:303–318
Kudela L (2013) Highly accurate subcell integration in the context of the finite cell method. Master’s thesis, Technical University Munich
Kudela L, Zander N, Bog T, Kollmannsberger S, Rank E (2015) Efficient and accurate numerical quadrature for immersed boundary methods. Adv Model Simul Eng Sci 2–10:1–22
Li S, Liu WK (2002) Meshfree and particle methods and their applications. Appl Mech Rev 55:1–34
Luo H, Pozrikidis C (2006) A Lobatto interpolation grid in the tetrahedron. IMA J Appl Math 71:298–313
Mirtich B (1996) Fast and accurate computation of polyhedral mass properties. J Graph Tools 1:31–50
Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 64:131–150
Mousavi SE, Sukumar N (2011) Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons. Comput Mech 47:535–554
Mousavi SE, Xiao H, Sukumar N (2010) Generalized Gaussian quadrature rules on arbitrary polygons. Int J Numer Methods Eng 82:99–113
Müller B, Kummer F, Oberlack M (2013) Highly accurate surface and volume integration on implicit domains by means of moment-fitting. Int J Numer Methods Eng 96:512–528
Ostachowicz W, Kudela P, Krawczuk M, \(\dot{\text{Z}}\) ak A (2011) Guided waves in structures for SHM: the time-domain spectral element method. Wiley, Hoboken
Parvizian J, Düster A, Rank E (2007) Finite cell method: h- and p-extension for embedded domain problems in solid mechanics. Comput Mech 41:121–133
Parvizian J, Düster A, Rank E (2012) Topology optimization using the finite cell method. Optim Eng 13:57–78
Patera AT (1984) A spectral element method for fluid dynamics: Laminar flow in a channel expansion. J Comput Phys 54:468–488
Persson PO (2005) Mesh generation for implicit geometries. PhD thesis, Massachusetts Institute of Technology
Pozrikidis C (2005) Introduction to finite and spectral methods using MATLAB. Chapman and Hall, Boca Raton
Ranjbar M, Mashayekhi M, Parvizian J, Düster A, Rank E (2014) Using the finite cell method to predict crack initiation in ductile materials. Comput Mater Sci 82:427–434
Rathod HT, Govinda Rao HS (1995) Integration of polynomials over linear polyhedra in Euclidean three-dimensional space. Comput Methods Appl Mech Eng 126:373–392
Ruess M, Tal D, Trabelsi N, Yosibash Z, Rank E (2012) The finite cell method for bone simulations: verification and validation. Biomech Model Mechanobiol 11:425–437
Ruess M, Schillinger D, Bazilevs Y, Varduhn V, Rank E (2013) Weakly enforced essential boundary conditions for NURBS-embedded and trimmed NURBS geometries on the basis of the finite cell method. Int J Numer Methods Eng 95:811–846
Schillinger D, Ruess M (2015) The finite cell method: a review in the context of high-order structural analysis of CAD and image-based geometric models. Arch Comput Methods Eng 22:391–455
Schillinger D, Düster A, Rank E (2012) The hp-d-adaptive finite cell method for geometrically nonlinear problems of solid mechanics. Int J Numer Methods Eng 89:1171–1202
Schillinger D, Ruess M, Zander N, Bazilevs Y, Düster A, Rank E (2012) Small and large deformation analysis with the p- and B-spline versions of the finite cell method. Comput Mech 50:445–478
Schillinger D, Cai Q, Mundani RP, Rank E (2013) A Review of the Finite Cell Method for Nonlinear Structural Analysis of Complex CAD and Image-Based Geometric Models. In: Advanced computing lecture notes in computational science and engineering, vol 93. Springer, Dordrecht, pp 1–23
Sehlhorst HG (2011) Numerical homogenization strategies for cellular materials with applications in structural mechanics. VDI Fortschritt-Berichte Reihe 18 Nr. 333
Sommariva A, Vianello M (2007) Product Gauss cubature over polygons based on Green’s integration formula. BIT Numer Math 47:441–453
Sudhakar Y, Moitinho de Almeida JP, Wall WA (2014) An accurate, robust, and easy-to-implement method for integration over arbitrary polyhedra: application to embedded interface methods. J Comput Phys 273:393–415
Szabó B, Babuška I (1991) Finite element analysis. Wiley, New York
Szabó B, Babuška I (2011) Introduction to finite element analysis: formulation, verification and validation. Wiley, New York
Talischi C, Paulino GH, Pereira A, Menezes IFM (2012) PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab. Struct Multidiscip Optim 45:309–328
Trimmer HG, Stern JM (1980) Computation of global geometric properties of solid objects. Comput Aided Des 12:301–304
Varduhn V, Hsu MC, Ruess M, Schillinger D (2016) The tetrahedral finite cell method: high-order immersogeometric analysis on adaptive non-boundary-fitted meshes. Int J Numer Methods Eng online:1–26
Verhoosel CV, van Zwieten G J, van Rietbergen B, de Borst R (2015) Image-based goal-oriented adaptive isogeometric analysis with application to the micro-mechanical modeling of trabecular bone. Comput Methods Appl Mech Eng 284:138–164
Wassermann B, Kollmannsberger S, Bog T, Rank E (2017) From geometric design to numerical analysis: a direct approach using the finite cell method on constructive solid geometry. Comput Math Appl 24. doi:https://doi.org/10.1016/j.camwa.2017.01.027 [Online]
Wassouf Z (2010) Die Mortar Methode für Finite Elemente hoher Ordnung. PhD thesis, Technical University Munich
Wenisch P, Wenisch O (2004) Fast octree-based voxelization of 3d boundary representation-objects. Tech. rep., Technical University Munich
Xu F, Schillinger D, Kamensky D, Varduhn V, Wang C, Hsu MC (2015) The tetrahedral finite cell method for fluids: immersogeometric analysis of turbulent flow around complex geometries. Comput Fluids online:1–20
Yang Z (2011) The finite cell method for geometry-based structural simulation. PhD thesis, Technical University Munich
Yang Z, Kollmannsberger S, Düster A, Ruess M, Grande Garcia E, Burgkart R, Rank E (2011) Non-standard bone simulation: interactive numerical analysis by computational steering. Comput Vis Sci 14:207–216
Yang Z, Ruess M, Kollmannsberger S, Düster A, Rank E (2012) An efficient integration technique for the voxel-based finite cell method. Int J Numer Methods Eng 91:457–471
Zander N, Kollmannsberger S, Ruess M, Yosibash Z, Rank E (2012) The finite cell method for linear thermoelasticity. Comput Math Appl 64:3527–3541
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Duczek, S., Gabbert, U. (2018). The Finite Cell Method: A Higher Order Fictitious Domain Approach for Wave Propagation Analysis in Heterogeneous Structures. In: Lammering, R., Gabbert, U., Sinapius, M., Schuster, T., Wierach, P. (eds) Lamb-Wave Based Structural Health Monitoring in Polymer Composites. Research Topics in Aerospace. Springer, Cham. https://doi.org/10.1007/978-3-319-49715-0_9
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