Assistant Professor of Applied Mathematics
Department of Mathematics, Lehigh University
I am currently an Assistant Professor at Department of Mathematics, Lehigh University.
My research concerns topics in the areas of numerical analysis, scientific computing and high performance computing, where I work on the development of novel numerical tools for models with background in science, engineering and biomedicine. I am particularly interested in applying the mathematical analysis knowledges in the design and analysis of efficient and stable numerical schemes for multiscale/multiphysics problems.
Recently, I have been working on developing a systematic unifying mathematical framework and the corresponding computational tool coupling the blood flows and the damaged soft tissues, with the target to improve the existing knowledge on the mechanism of brain aneurysm rupture and prosthetic heart valves failure. Moreover, stability and accuracy issues appear while developing the coupling strategies for this multiscale/multiphysics framework, and I develop novel mathematical analysis to guide the model/algorithm designs.
Specifically, my research interests include:
Email: yac310 [at] lehigh [dot] edu
Ph.D. candidate in Mechanical Engineering & Mechanics (Co-advising with Dr. Alparslan Oztekin)
Research Interests: Lattice Boltzmann Method, Computational Fluid Dynamics
Email: huy316 [at] lehigh [dot] edu
Ph.D. candidate in Applied Mathematics
Research Interests: Peridynamic Models, Numerical Analysis
Email: yun316 [at] lehigh [dot] edu
Graduate Student in Applied Mathematics
Research Interests: TBD
Email: kaw418 [at] lehigh [dot] edu
Undergraduate Student in Applied Mathematics
Research Topic: Numerical Methods for Coupling Peridynamics and Classical Theory
B.Sc. in Applied Mathematics, Lehigh University
Master Student of Statistical Practice at Carnegie Mellon University
M.Sc. in Applied Mathematics, Lehigh University
Assistant Professor of Mathematics in Northampton Community College
Undergraduate Student in Biological Science, Northampton Community College
Awarded as Academic All-Star and member of the All-PA Academic Team by the international honor society Phi Theta Kappa and the Pennsylvania Commission for Community Colleges
Mixed Spectral/hp Element Formulation for Nonlinear Elasticity: To model human arterial walls, nearly incompressible nonlinear elasticity is widely employed. However, numerical solvers is not accurate, especially for models with very high bulk modulus (means the material is very incompressible). This is a phenomenon called ``volumetric locking" behavior. Two main strategies were then introduced for overcoming volumetric locking: the displacement-only formulation with high-order spectral elements and the modified variational forms, including the mixed formulations. However, for the first strategy one has to increase greatly the computational cost to resolve the volumetric locking problem. On the other hand, the mixed formulation does not have this problem, but it complicates the theoretical framework of the method as the element spaces must be compatible.
We developed the pressure/displacement mixed formulation to high-order spectral/hp elements, and investigated the two types of locking phenomena and stability conditions. Compared to the displacement-only formulation, the mixed formulation has superior accuracy and is overall more efficient, especially for problems with high bulk modulus. In Figure 1 we show the volumetric locking test on a thick-walled tube under pressure load Pinner = 0.5 MPa (applied along the radial direction). We test both mixed (red lines) and displacement-only formulations (blue lines) for this problem, with bulk modulus k from 1 to 100000. The results from different element spaces for pressure (with order Op) and displacement (with order Ou) are investigated, demonstrated that using low-order elements for displacement, the displacement-only formulation suffers from volumetric locking problem, while the mixed formulation does not.
Our study also contributed to the understanding the stability condition and the compatible element spaces: we discovered the most robust choice of the mixed formulation as Ou=Op+1, leading to stable and accurate results.
A Semi-Local Spectral/hp Element Solver for Elasticity Problems: In the past decades, several of schemes and preconditioners have been developed to accelerate parallel high-order fluid solver, but few efforts were devoted to structure solver. While this issue may have been resolved for low-order finite elements it is a great challenge for spectral and high-order elements. In many fluid-structure interaction simulations, most of the CPU time is consumed by solving the linear systems arising in the structure part, especially for large-scale cases such as patient specific aneurysm problems. In general, this problem is more severe when high order method is using because the high order method requires more operations in the iterative solver for the linear systems.
We developed an efficient semi-local method for speeding up the solution of the linear systems of the high-order structure solver on parallel computers. The main idea is to divide the computation loads into small regions (which is called elements) and then approximate the element-wise residual distribution with a carefully designed operator, and subsequently solve smaller linear systems on each small regions (element). Because almost all the operations are treated element-wise, we obtained good parallel scalability and substantial speed-up, especially for fluid-structure interaction simulations. The method has also been successfully applied on patient-specific aneurysm fluid-structure interaction simulations, as in the left plot of Figure 2. The averaged CPU time is compared in the middle and the right plots, where we also show the speed-up factor from the semi-local method in each subiteration. When using polynomial order larger than 2, the semi-local method is more efficient than the original spectral/hp element formulation, and while employing elements with polynomial order 5, the speed-up factor can be as high as 4.5. Therefore, the best speed-up is achieved for fluid-structure interaction tests on high order elements.
Fractional Modeling of Viscoelasticity in 3D Cerebral Arteries and Aneurysms: Typically, constitutive laws for arterial walls are derived using integer-order differential equations that model stress-strain relations. In these models, the viscous behaviors are very sensitive to the relaxation parameters. However, estimation of these anatomic location-specific parameters remains a challenging task in clinical practice.
We have studied 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 employed a fast convolution method that reduces the memory cost to O(log(N)) and the computational complexity to O(N\log(N)). Furthermore, we combined the fast convolution with high-order backward differentiation to achieve high-order time integration accuracy. As shown in Figure 3, when combining with a 3rd order BDF scheme, 3rd order accuracy in time is achieved, and O(log(N)) computational complexity is observed for each time step.
We confirmed 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 applied the method to modeling fluid-structure interaction of a 3D patient-specific compliant cerebral artery with an aneurysm, with its parameters obtained from the loading test of a brain aneurysm sample from [Robertson et al., 2015]. The simulation results suggest a phase delay in the time traces of displacement magnitudes, and the displacement magnitude distribution patterns from this model has predicted the largest deformed location at the aneurysm pole, where rupture usually occurs.
Multiscale models for soft tissue damage: Currently, most of the numerical simulations for aneurysms are performed with continuum based models, but the macroscopic phenomenon of rupture is a result of physical processes with their origin in the atomistic scale. Therefore, developing a mathematical framework which could capture multiscale phenomena would be valuable for studying cerebral aneurysm rupture.
The peridynamic theory is a spatially nonlocal formulation which unifies the modeling of continuum media and discontinuities, and therefore the material damage can be captured autonomously as a natural component of the material deformation. For instance, in the crack propagating problem as shown in Figure 4. The peridynamic theory also has a parallel to both the molecular dynamics computations and the classical constitutive model. Therefore, the peridynamic model can simulate a wide range of complex problems in fracture and damage mechanics and can be consistently coupled with its corresponding classical constitutive model in the undamaged area. These two properties make the peridynamic theory especially suitable for the multiscale modeling framework.
In this project, we employ the peridynamics model to capture the material failure. In the region where material failure is expected to initiate, a discretized particle solver based on the peridynamic theory is employed. In the rest of the problem domain, the material is modeled by the classical elastic theory, and numerically discretized with the finite element method. The two models are coupled in a concurrent way, as illustrated in the left plot of Figure 5. For quasi-static problems, we employ iterations in the concurrent coupling scheme, and investigate different coupling strategies as well as interface boundary conditions. The numerical experiments suggest the Robin boundary condition together with the Aitken algorithm for its robustness and efficiency (as shown in the right plot of Figure 5).
Generalized Fictitious Methods for FSI in Hemodynamic Problems: In hemodynamic problems, the arterial wall density is relatively light and usually close to the blood density. However, this brings a special challenge for numerical simulations: the convergence of the partitioned FSI procedure is problematic because of the so-called added-mass effect.
We developed a family of fictitious methods for FSI problems, where additional terms were introduced in the fluid/structure equations to balance the added-mass effect, which makes the partitioned FSI solver perform closer to an exact coupled solver. Moreover, this method can be easily extended and implemented in any existing fluid-structure interaction solver. Only one side of the solver needs to be implemented. One could combine a black-box fluid solver with an open-source structure solver, or vice verse. This feature offers more flexibility to the method, and it is especially important for commercial use. We also provided mathematical analysis to obtain optimal values for the fictitious parameters as well as to reveal a similarity with the popular Robin-based approach. In Figure 6, simulation results for 3D patient-specific brain arteries with aneurysm are presented, as a validation for the methods on complex geometries.
Fictitious Inertia Method for FSI in Oil Industry Problems: We have further extended the fictitious method to applications of deepwater offshore oil operations. In the oil industry, when the sea current is strong, the oil pipes under water vibrates with the fluid vortex and such a vibration would damage the oil pipes and even cause the rupture. This phenomenon is called the vortex-induced vibrations (VIV). Therefore, researches were devoted on developing the vortex-induced vibrations suppressors which attach to the oil pipes, change the fluid vortex and help to reduce the vibrations. Generally, the vortex-induced vibrations (VIV) suppressors employed in industry are nearly-neutrally buoyant, therefore with low mass and low rotational inertia. This makes the numerical simulations difficult, and commercial software are inadequate because of the added-mass effect.
In this project, with the fictitious method we build the model for fairing, one kind of popular VIV suppressors, as displayed in Figure 7, and quantified numerically for the first time various salient features of 2D and 3D free-to-rotate devices for VIV suppression and related them to modified flow structures in the near wake (as shown in Figure 8). The fictitious method has been deployed in the commercial software for industrial use. This work helped the development of VIV supressors in the oil industry.
Immersogeometric Methods for Heart Valve Simulations: Fluid--structure interaction models of heart valves present a number of unique challenges:
We have worked on the heart valve problem regularity and numerical verification for immersogeometric methods. When the heart valves are closed, the fluid pressure is not continuous across the thin valves, and the gradient of fluid velocity in the normal direction has a jump across the heart valves. So the solution of velocity has degraded smoothness in the heart valve problems, and this solution regularity should be addressed for the error estimates since it may constrain the accuracy of numerical methods. We formulated the immersogeometric coupling strategies in a parabolic model problem, when showed that with the assumption that the fluid subdomain boundary is smooth, the solution has regularity for any >0. With this regularity result, when the mesh size is h and element size for the Lagrange multiplier H>h is large enough, we proved the error estimates for the immersogeometric method on this parabolic model problem:
To validate the immersogeometric method in vitro, we have further compared the simulation results with experimental results on a latex valve in an acrylic tube. In Figure 9, the computed valve deformations are compared with the corresponding images collected in the experiment at four different time points. At each time point, the upper right plot shows the experiment image, while the lower right plot shows the simulated deformation. The left plot at each time point shows the contour plots of fluid velocity magnitudes from computation on slices, which illustrates the ability of computer simulations to provide additional information about the flow field and the full 3D deformation of the leaflets that are difficult to measure experimentally.
The HPC workshop is a high-performance computing (HPC) workshop sponsored by the National Science Foundation, the Lehigh ADVANCE program, and the Department of Mathematics, for students, faculty members from the Lehigh Valley Association of Independent Colleges (LVAIC) schools and the general public in Lehigh Valley area. The workshop aims to promote female researchers in scientific computing therefore would prefer female attendees, but male attendees are also welcome. Besides tutorials and hands-on exercises, researchers working on high-performance computing in both academic and industry will also present in the workshop, so the attendees will have opportunities to get connected with these researchers.
Dr. Yu and Dr. Pacheco host the workshop annually, typically near the end of the spring semester. The 2017 HPC workshop took place on 31 May-1 June, 2017. For the 2018 HPC workshop, further details will be posted during the 2017-2018 spring semester.
Contact Email: yuy214 [at] lehigh [dot] edu
Last modified on August 1, 2017 | Copyright © by Yue Yu. All rights reserved.