ACME 2011 Proceedings of the 19 UK National Conference of the ...

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3.2 Twisting test In order to verify that the fibres are correctly implemented, the disc is subjected to a torque θ = 3 ◦ , with both endplates enforced to remain parallel. The deformations are therefore proportional to the distance from the centre of rotation, which is reflected in Fig. 4a for the fibres experiencing tension. The other family of fibres experiencing compression is not active. (a) Family of fibres in tension (b) Family of fibres in compression Figure 4: Fibre stress contour plot in the mid-sagital plane for the twisted disc 4 CONCLUSION A 3D biphasic swelling, anisotropic and hyperelastic model has been formulated. It captures the salient features of the intervertebral disc such as ionic, stiffening and swelling effects and phase coupling. The early stage results show qualitatively good results. Work is under progress to utilise more realistic geometries and define physiologically correct initial conditions. References [1] R. Eberlein, G.A. Holzapfel, and C. A. J. Schulze-Bauer. An anisotropic model for annulus tissue and enhanced finite element analyses of intact lumbar disc bodies. Computer Methods in Biomechanics and Biomedical Engineering, 4(3):209–229, 2001. [2] W. Ehlers, N. Karajan, and B. Markert. An extended biphasic model for charged hydrated tissues with application to the intervertebral disc. Biomechanics and Modeling in Mechanobiology, 8(8):1617–7959, 2008. [3] P. Heneghan and P. E. Riches. The strain-dependent osmotic pressure and stiffness of the bovine nucleus pulposus apportioned into ionic and non-ionic contributors. Journal of Biomechanics, 41(11):2411–2416, 2008. [4] G. A. Holzapfel, C. A. J. Schulze-Bauer, G. Feigl, and P. Regitnig. Single lamellar mechanics of the human lumbar anulus fibrosus. Biomechanics and Modeling in Mechanobiology, 3(3):125–140, 2005. [5] W. M. Lai, J. S. Hou, and V. C. Mow. A triphasic theory for the swelling and deformation behaviors of articular-cartilage. Journal Of Biomechanical Engineering-Transactions Of The Asme, 113(3):245–258, 1991. 28
Proceedings of the 19 th UK Conference of the Association for Computational Mechanics in Engineering 5 – 6 April 2011, Heriot-Watt University, Edinburgh A ONE-DIMENSIONAL BLOOD FLOW MODEL FOR STUDYING THE EFFECT OF AORTIC ANEURYSMS IN HUMAN ARTERIAL NETWORK Kenny Low*, Perumal Nithiarasu and Igor Sazonov Civil and Computational Engineering Centre, College of Engineering Swansea University, Singleton Park, Swansea SA2 8PP, UK *Corresponding author: 400727@swansea.ac.uk Key Words: blood flow; fusiform aortic aneurysm; one-dimensional; locally conservative Galerkin; flow waveforms ABSTRACT The effect of including a fusiform aortic aneurysm into an arterial network is studied and the differences in the waveforms are identified. An existing one-dimensional (1D) model is modified to include the aneurysms and the waveforms are computed using a realistic ventricular pressure. The 1D fluid flow equations are solved using the locally conservative Galerkin (LCG) finite element method. This method provides an explicit element-by-element conservation and naturally incorporates vessel branching, where arteries with different discontinuities are permitted to share the same location at the start of the branch. An example of fusiform thoracic aortic aneurym with three different sizes, but remain in the same location, is studied and the flow waveforms are compared against a normal adult waveform. 1 INTRODUCTION Aortic aneurysm is generally a balloon-like bulge formed through weakened arterial wall of the aorta and can be of saccular or fusiform in nature. Fusiforms can be found in the upper or the abdominal part of the aorta. An aneurysm at the upper part is referred to as Thoracic Aortic Aneurysms (TAA) and the one at the lower part is known as Abdominal Aortic Aneurysms (AAA). Aortic aneurysm incidents have steadily increased over the past few decades. Nonetheless, there are still no known precise reasons which affect the weakening of the aortic wall. However, many believed that a number of factors have been related to this disease [1]. Aneurysms often cause no symptoms and are rarely noticed before rupture. To overcome this crisis, aneurysms need to be detected at an early stage [1, 2]. Some studies have been carried out by [2] to investigate the effect of an AAA on the wave propagation both experimentally and using a 1D arterial network model. Nevertheless, if this effect can be evaluated through common pulse sites such as common carotid artery, radial artery and many more, there might be an opportunity for diagnosing aneurysms. The 1D arterial network models have been used extensively in recent years, primarily to provide a better understanding of the blood flow in the arterial tree. They offer a good compromise between accuracy and cost (computational time) [3]. In this paper, the existing 1D model of [4] with an additional fusiform aneurysm embedded is used to compute the waveforms in the human arterial system using a realistic ventricular pressure as an input. The main objective is to study the effect of blood flow in the presence of an aortic aneurysm in a human arterial network and identify the differences in waveforms. The governing equations are solved using the LCG finite element method. This method provides explicit element-by-element conservation and reintroduces an interface flux in each element. The element-byelement solution, which is independent of the surrounding elements, offers controlled flexibility thus allowing the introduction of discontinuities along the element boundary [5, 6]. This property is useful in the consideration of vessel branching points such as bifurcations or trifurcations. Arteries with different 29
Page 1: ACME 2011 Proceedings of the 19 th
Page 4 and 5: First published 2011 by Heriot-Watt
Page 7 and 8: CONTENTS Invited Lectures CHALLENGE
Page 9 and 10: A DAMAGE-MECHANICS-BASED RATE-DEPEN
Page 11: A PROJECTED QUASI-NEWTON METHOD FOR
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Practicallly, the perturbation fiel
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Proceedings of the 19 th UK Confere
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3 SIMULATING CONCRETE MESO-STRUCTUR
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¯K i is an approximation of the co
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h = 5.36 matrix 4.3 z P E f νf 0.9
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From the total hoop strainε θ , t
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3 EXTERIOR POINT ESHELBY BASED CRAC
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is applied uniformly on the edge to
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(a) (b) (c) (d) Figure 1: Influence
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On each subdomain, we then have to
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( i�1) ( i�1) �Un�1 � ε
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� ~ � 1 ψ ~ � � ~ (2) �
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For the case of uni-axial tension,
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3 Numerical results Preliminary num
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The coefficients An cannot be deriv
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The spatial semi-discretisation is
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3 CRITICAL TIME STEP INCREASED WITH
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v/vo 2E-05 1.5E-05 1E-05 5E-06 0 -3
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3 RESULTS AND DISCUSSION Fig. 1a sh
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Model updating from eigenvalue and
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1. If R = ω 2 n, there is no eigen
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F = ´ Ω STv CSv dΩ A = ´ Γ U
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Fig. 3 Wall normal pressure in the
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contact contact (ii). f ≈ f ( d,
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In the analysis of the incident P-w
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elow a certain threshold. In both t
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K/d 2 Effective Permeability 0.0032
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METHOD 2 (Mellor and Herring Type)
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(a) Free Surface Profile from Janos
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2 IMMERSED FLUID SOLVER Consider th
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References [1] R. Mittal and G. Iac
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a second impact on the forward fuse
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Acknowledgement L. Papagiannis is f
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acceptable probability throughout t
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algorithm was then responsible for
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strategy of saving the computationa
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(a) (b) Figure 5. Comparison of (a)
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Thermal load step i i i i Update ,
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Displacement (mm) Conclusions 45 40
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load carrying of the structure to b
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compressive stress field between co
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Unbonded flexible pipes and risers
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Figure 2: (a) Riser configuration (
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Furthermore the total strain varies
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Displacement 14 12 10 8 6 4 2 0 Mem
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As this project was in the conceptu
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kg and additional mass to represent
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1 Step make box 2 Step make cylinde
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element method. The solid model is
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just new element classes) without t
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Fig.4. Frame model and the displace
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capacities to meet with such demand
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esponses. Severe cases are to be co
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2 Numerical results In this section
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the frequency intially increases un
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and F (R) = ⎡ ⎢ ⎣ ... UαT (R
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Merit function g 10 1 10 0 10 -1 10
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2 STRUCTURAL CHARACTERISTICS OF PTM
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Fig. 5. Simplified FE model of a PT
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general the computational results o
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References [1] Ainsworth M, Oden JT
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Error Cause Interpolation Error Ele
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(a) Before Improvement (b) After Im
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een applied to FE fluid flow proble
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Acknowledgements Figure 2: Flowchar
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1 1 ′′ 1 ′′′ 3 ′′ 2
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3 COMBINING SHAPE FUNCTIONS The exp
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electron current). For the case of
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electron concentration (1/cm 3 ) x
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2 CONSERVATION LAW FOR A MOVING CV
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5 NUMERICAL RESULTS A 1-D cellular
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FEM can be found in [3]. In this pa
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h Y 1 0 −1 −2 −3 −4 −5
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applications. The partition of unit
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1.1 1.05 1 0.95 0.9 0.85 0.8 0.75 2
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2 NURBS BASIS FUNCTIONS NURBS are a
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4 RESULTS To illustrate the isoBEM
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290
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1 1 2 { σ ( ξ, s) } = [ D] { ε (
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4.4 Example 4 - A single edge crack
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