published_or_final_version / Mechanical Engineering / Doctoral / Doctor of Philosophy

published_or_final_version / Mechanical Engineering / Doctoral / Doctor of Philosophy

published_or_final_version / Civil Engineering / Doctoral / Doctor of Philosophy

published_or_final_version / Mechanical Engineering / Master / Master of Science in Engineering

published_or_final_version / abstract / toc / Civil Engineering / Master / Master of Philosophy

published_or_final_version / Civil Engineering / Master / Master of Philosophy

Thesis (Ph. D.)--University of Hong Kong, 2006. / Title proper from title frame. Also available in printed format.

In this thesis, numerical experiments are performed to test the numerical stability of the finite element method for analyzing incompressible materials from boundary displacements. The significance of the study relies on the fact that incompressibility, or density preservation during deformation, is an important property of materials such as rubber and soft tissue.<p>It is well known that the finite element analysis (FEA) of incompressible materials is less straightforward than for materials which are compressible. The FEA of incompressible materials using the usual displacement based finite element method results in an unstable solution for the stress field. Hence, a different formulation called the mixed u-p formulation (u displacement, p pressure) is used for the analysis. The u-p formulation results in a stable solution but only when the forces and/or stress tractions acting on the structure are known. There are, however, certain situations in the real world where the forces or stress tractions acting on the structure are unknown, but the deformation (i.e. displacements) due to the forces can be measured. One example is the stress analysis of soft tissues. High resolution images of initial and deformed states of a tissue can be used to obtain the displacements along the boundary. In such cases, the only inputs to the finite element method are the structural geometry, material properties, and boundary displacements. When finite element analysis of incompressible materials with displacement boundary conditions is performed, even the mixed u-p formulation results in highly unstable calculations of the stress field. Here, a hypothesis for solving this problem is developed and tested. Theories of linear and nonlinear stress analysis are reviewed to demonstrate that it may be possible to determine the von Mises stress uniquely in spite of the numerical instability inherent in the calculations.<p>To validate this concept, four different numerical examples representing different deformation processes are considered using ANSYS®: a plate in simple shear; expansion of a thick-walled cylinder; a plate in uniform strain; and Cooks membrane. Numerical results show that, unlike the normal stress components Sx, Sy, and Sz, the calculated values of the von Mises stress are reasonably accurate if measurement errors in the displacement data are small. As the measurement error increases, the error in the von Mises stress increases approximately linearly for linear problems, but can become unacceptably large in nonlinear cases, to the point where solution process encounter fatal errors. A quasi-Dirichlet patch test in association with this problem is also introduced.

In this thesis, numerical experiments are performed to test the numerical stability of the finite element method for analyzing incompressible materials from boundary displacements. The significance of the study relies on the fact that incompressibility, or density preservation during deformation, is an important property of materials such as rubber and soft tissue.<p>It is well known that the finite element analysis (FEA) of incompressible materials is less straightforward than for materials which are compressible. The FEA of incompressible materials using the usual displacement based finite element method results in an unstable solution for the stress field. Hence, a different formulation called the mixed u-p formulation (u displacement, p pressure) is used for the analysis. The u-p formulation results in a stable solution but only when the forces and/or stress tractions acting on the structure are known. There are, however, certain situations in the real world where the forces or stress tractions acting on the structure are unknown, but the deformation (i.e. displacements) due to the forces can be measured. One example is the stress analysis of soft tissues. High resolution images of initial and deformed states of a tissue can be used to obtain the displacements along the boundary. In such cases, the only inputs to the finite element method are the structural geometry, material properties, and boundary displacements. When finite element analysis of incompressible materials with displacement boundary conditions is performed, even the mixed u-p formulation results in highly unstable calculations of the stress field. Here, a hypothesis for solving this problem is developed and tested. Theories of linear and nonlinear stress analysis are reviewed to demonstrate that it may be possible to determine the von Mises stress uniquely in spite of the numerical instability inherent in the calculations.<p>To validate this concept, four different numerical examples representing different deformation processes are considered using ANSYS®: a plate in simple shear; expansion of a thick-walled cylinder; a plate in uniform strain; and Cooks membrane. Numerical results show that, unlike the normal stress components Sx, Sy, and Sz, the calculated values of the von Mises stress are reasonably accurate if measurement errors in the displacement data are small. As the measurement error increases, the error in the von Mises stress increases approximately linearly for linear problems, but can become unacceptably large in nonlinear cases, to the point where solution process encounter fatal errors. A quasi-Dirichlet patch test in association with this problem is also introduced.

Abdominal aortic aneurysms (AAA) are an expansion in diameter of the abdominal aorta and their rupture is a leading cause of mortality. One of the treatments for AAA is the implantation of an endograft (also called a stent graft), a combination of fabric and metal stents, to provide a new conduit for blood and shield the aneurysm sac from direct pressurization. After implantation of the stent graft, the aneurysm may shrink, grow, or stabilize in diameter ? even in the absence of apparent flow into the sac ? in some cases resulting in graft failure through component separation, kinking, or loss of seal at its ends. Greater understanding of AAA and treated AAA could provide insight on how treatment might be modified to improve treatment methods and/or design devices to be more effective in a wider range of patients. Computational models provide a means to investigate the biomechanics of endografts treating AAA through analysis of the endografts, the AAA, and the combination of them. Axisymmetric models of endograft-treated AAA showed that peak von Mises stress within the wall varied between 533 kPa and 1200 kPa when different material properties for the endograft were used. The patient-specific models, built from time series of patient CT scans with similar patient history but different outcomes, show that wall shrinkage and stability can be related to the level of stresses within the vessel wall, with the shrinking AAA showing a greater reduction by endograft treatment and a lower final value of average von Mises stress. The reduction in pressure felt by the wall is local to the central sac region. The inclusion of thrombus is also essential to accurate stress estimation. The combination of axisymmetric and patient-specific computational models explains in further detail the biomechanics of endograft treatment. The patient-specific reconstruction models show that when effectively deployed and reducing the pressure felt in the AAA wall, the graft is under tension in the sac region and compression at its ends.