Modeling stent-type structures using geometrically exact beam theory Nora Hagmeyer, Ivo Steinbrecher, Alexander Popp University of the Bundeswehr Munich, Institute for Mathematics and Computer-Based Simulation October 24th 2018
Institute for Mathematics and Computer- Based Simulation Overview Motivation Abdominal Aortic Aneurysm(AAA) Endovascular Aortic Repair(EVAR) Bottom-up modeling approach Beam interaction frameworks Beam-to-solid meshtying Beam-to-solid contact Beam-to-beam contact Fluid-beam interaction Outlook Nora Hagmeyer| Modeling stent-type structures using geometrically exact beam theory 2
Institute for Mathematics and Computer- Based Simulation Abdominal Aortic Aneurysm(AAA) • AAA = ballooning of the aorta (diameter > 3.5 cm) in abdominal region • catastrophic rupture in 25% of all cases, rupture mortality 90% • epidemiology (Germany): � 200,000 people affected � 15,000 surgeries/year Figure: c � Vascular Surgery @ TUM • surgical repair � conventional open repair � endovascular repair (EVAR) 30-day peri-operative mortality rate • Open repair: 4,7% • EVAR: 1,9% [Greenhalgh et al., Lancet, 2004] Nora Hagmeyer| Modeling stent-type structures using geometrically exact beam theory 3
Institute for Mathematics and Computer- Based Simulation Endovascular Aortic Repair(EVAR) • still early ( 4%) and late ( 15%) major complications � stent graft migration � endoleakage (type I / type II) � damage of arterial wall � secondary AAA rupture possible • success of EVAR is still limited by these risks • dif fi cult to predict – even for experienced surgeons • availability of experimental / clinical data is limited • EVAR process is largely dominated by arterial stent graft Figure: c � Vascular Surgery @ TUM [Lin et al., J Vasc Surg, 2011] Nora Hagmeyer| Modeling stent-type structures using geometrically exact beam theory 4
Institute for Mathematics and Computer- Based Simulation Bottom-up modeling approach Complex microstructure of stents and stent grafts • stent wire shapes: helix, sine waves, . . . • graft topology: straight, curved, bifurcated, . . . × • stent-graft connection: sutures, bonding, pre-stressed, . . . • materials and boundary conditions: anisotropy, inelasticity, crimping, . . . [Demanget et al., J Endovasc Ther, 2013] Nora Hagmeyer| Modeling stent-type structures using geometrically exact beam theory 5
Institute for Mathematics and Computer- Based Simulation Modeling stent structures using reduced order models Equilibrium equations (Simo-Reissner beam theory) f � + ˜ f = ˙ L t m � + r � × f + ˜ m = ˙ H t Figure: 100 volume elements • geometrically exact (GE) beam element formulations: (3D) vs. 1 beam element (1D) known for their high accuracy and computational ef fi ciency [Romero, 2008], [Bauchau et al., 2014] • Simo-Reissner theory of thick rods (6 modes: axial tension, 2x shear, torsion, 2x bending) [Reissner, 1972], [Reissner, 1981], [Simo, 1985] • Kirchhoff-Love theory of thin rods Figure: Simo-Reissner beam (4 modes: axial tension, torsion, 2x bending) centerline [Kirchhoff, 1859], [Love, 1944] Nora Hagmeyer| Modeling stent-type structures using geometrically exact beam theory 6
Institute for Mathematics and Computer- Based Simulation Beam interaction frameworks Beam-to-solid Beam-to-beam meshtying contact e.g. stent and graft e.g. stent crimping bonding in catheter Fluid-beam Beam-to-solid interaction contact e.g. stent e.g. stent placement in artery placement in artery Nora Hagmeyer| Modeling stent-type structures using geometrically exact beam theory 7
Institute for Mathematics and Computer- Based Simulation Beam-to-solid meshtying Beam-to-solid meshtying reaction force[N] Enforcement of the Meshtying constraint � 1 � r ( 1 ) − ˆ x ( 2 ) � � r ( 1 ) − ˆ x ( 2 ) � W mt = 2 ε ds Γ ( 1 ) mt or � � r ( 1 ) − ˆ x ( 2 ) � W LM = λ bending angle [ ◦ ] ds Γ ( 1 ) mt Figure: Experimental setup at University of Tokyo and Shibaura Institute of Technology • simple bending of initially straight stent graft • characteristic load case during stent graft lifetime • realistic geometrical and material parameters Nora Hagmeyer| Modeling stent-type structures using geometrically exact beam theory 8
Institute for Mathematics and Computer- Based Simulation Beam-to-solid contact • contact introduces nonlinearity • inequality constraints require additional consideration Beam-to-solid contact KKT-type conditions modeling contact mechanics g n ( X , t ) ≥ 0 Figure: Projection p n ( X , t ) ≤ 0 p n ( X , t ) · g n ( X , t ) = 0 Nora Hagmeyer| Modeling stent-type structures using geometrically exact beam theory 9
Institute for Mathematics and Computer- Based Simulation Beam-to-beam contact Beam-to-beam contact Contact constraint formulations W c , point = 1 2 ε � g � 2 or � l 1 1 2 ε � g ( ξ ) � 2 ds 1 W c , line = Γ 0 • point-to-point contact formulation • line-to-line contact formulation • all-angles beam contact (ABC) formulation a) Point-to-point contact formulation b) Line-to-line contact formulation Nora Hagmeyer| Modeling stent-type structures using geometrically exact beam theory 10
Institute for Mathematics and Computer- Based Simulation Fluid-beam interaction • Immersed Boundary type method FSI forces • large movements possible using the fl uid shape functions to calculate • possibility to model interaction between the forces acting on the beam blood and arterial fl uid via ALE approach • possibility of integrating beam-wall contact mechanics x f , 3 x f , 4 x f , 3 x f , 4 x beam x beam x f , 1 x f , 2 x f , 1 x f , 2 Fluid-beam interaction a) FSI forces b) Velocity "spreading" a NS ( v , p ; ∂ v , δ p ) = F f ( δ v , δ p ) a beam ( r ( s ) , Λ ( s ) ; δ r , δθ ) = F beam () Velocity � ˆ s � f � ds spreading operator S x f to transfer the beam σ ( v , p ) n ds = velocities to the adjacent fl uid nodes using Γ beam 0 r ( s ) v beam = v f ˙ the fl uid shape functions Nora Hagmeyer| Modeling stent-type structures using geometrically exact beam theory 11
Institute for Mathematics and Computer- Based Simulation Outlook • Implementation of the Fluid-beam interaction framework • Convergence studies and comparison to full 3D simulations • Comparison of different constraint enforcement techniques • Segmentation of integration domain • Coupling of the rotational DOFs of the beam • Combining the different frameworks Nora Hagmeyer| Modeling stent-type structures using geometrically exact beam theory 12
Institute for Mathematics and Computer- Based Simulation Thank you for your attention Nora Hagmeyer| Modeling stent-type structures using geometrically exact beam theory 13
Institute for Mathematics and Computer- Based Simulation References Andreas D Rauch, Anh-Tu Vuong, Lena Yoshihara, and Wolfgang Wall. A coupled approach for fl uid saturated poroelastic media and immersed solids for modeling cell-tissue interactions. International journal for numerical methods in biomedical engineering , page e3139, 08 2018. Christoph Anton Meier. Geometrically exact fi nite element formulations for slender beams and their contact interaction . PhD thesis, Technische Universität München, 2016. Charles S Peskin. The immersed boundary method. Acta numerica , 11:479–517, 2002. Alexander Popp. Mortar methods for computational contact mechanics and general interface problems . PhD thesis, Technische Universität München, 2012. Nora Hagmeyer| Modeling stent-type structures using geometrically exact beam theory 14
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