Fractional modeling of viscoelasticity in 3D cerebral arteries and aneurysms

详细信息    查看全文

• 作者：Yue Yu^a ; ^{yuy214@lehigh.edu" class="auth_mail" title="E-mail the corresponding author} ; Paris Perdikaris^b ; ^{parisp@mit.edu" class="auth_mail" title="E-mail the corresponding author} ; George Em Karniadakis^c ; ^{gk@dam.brown.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Fast convolution method ; Fractional differential equations ; Fluid&ndash ; structure interaction ; Spectral element method ; Patient-specific vasculature ; Brain aneurysm
• 刊名：Journal of Computational Physics
• 出版年：2016
• 出版时间：15 October 2016
• 年：2016
• 卷：323
• 期：Complete
• 页码：219-242
• 全文大小：2317 K

文摘

We develop efficient numerical methods for fractional order PDEs, and employ them to investigate viscoelastic constitutive laws for arterial wall mechanics. Recent simulations using one-dimensional models [1] have indicated that fractional order models may offer a more powerful alternative for modeling the arterial wall response, exhibiting reduced sensitivity to parametric uncertainties compared with the integer-calculus-based models. Here, we study three-dimensional (3D) fractional PDEs that naturally model the continuous relaxation properties of soft tissue, and for the first time employ them to simulate flow structure interactions for patient-specific brain aneurysms. To deal with the high memory requirements and in order to accelerate the numerical evaluation of hereditary integrals, we employ a fast convolution method [2] that reduces the memory cost to formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021999116302637&_mathId=si1.gif&_user=111111111&_pii=S0021999116302637&_rdoc=1&_issn=00219991&md5=92b3b81271a069e27b7ddc2c13c5b67e" title="Click to view the MathML source">O(log⁡(N))flow="scroll">Ofalse">(log⁡false">(Nfalse">)false">) and the computational complexity to formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021999116302637&_mathId=si2.gif&_user=111111111&_pii=S0021999116302637&_rdoc=1&_issn=00219991&md5=9bae570b6ad07a683ee14fffb9cc6ccf" title="Click to view the MathML source">O(Nlog⁡(N))flow="scroll">Ofalse">(Nlog⁡false">(Nfalse">)false">). Furthermore, we combine the fast convolution with high-order backward differentiation to achieve third-order time integration accuracy. We confirm that in 3D viscoelastic simulations, the integer order models strongly depends on the relaxation parameters, while the fractional order models are less sensitive. As an application to long-time simulations in complex geometries, we also apply the method to modeling fluid–structure interaction of a 3D patient-specific compliant cerebral artery with an aneurysm. Taken together, our findings demonstrate that fractional calculus can be employed effectively in modeling complex behavior of materials in realistic 3D time-dependent problems if properly designed efficient algorithms are employed to overcome the extra memory requirements and computational complexity associated with the non-local character of fractional derivatives.