A matrix-free iterative solution strategy has been developed for implicit immersed continuum methods. Mixed finite element formulations are employed for both the immersed compressible solid and the compressible surrounding fluid.

In linear acoustoelastic analysis, it has been widely reported that the displacement-based fluid elements employed in frequency or dynamic analyses exhibit spurious non-zero frequency circulation modes (Kiefling, 1976; Hamdi, et al., 1978; and Olson and Bathe, 1983). Various approaches have been introduced to obtain improved formulations, including a 4-node element based on a reduced integration technique (Chen and Taylor, 1990), the displacement potential and pressure formulation (Morand and Ohayon, 1979), and the velocity potential formulation (Everstine, 1981; Olson and Bathe, 1985; and Felippa and Ohayon, 1990). The mixed displacement/pressure finite element formulation originally proposed by Wang and Bathe, 1997, has been proven to be reliable and free of spurious nonzero frequencies. A recent mathematical study of this formulation was also published in Bao, Wang, and Bathe, 2001. In this paper, mixed finite element formulations are extended to the study of cochlea, in particular, the resonant frequency of the enclosed cavity with different geometries. In essence, immersed flexible structures with and without opening are compared at different physical parameters such as variable thickness, fluid and solid densities.

Deformable flexible structures immersed in aqueous environment are ubiquitous in biological systems. In addition to large structural deformations and nonlinear material behaviors, complex chemical and physical conditions at micro-scale also play important roles in overall system behaviors. In immersed boundary/continuum methods, independent solid meshes move on top of a fixed or prescribed background fluid mesh. This simple stragtegy enables a natural coupling of various immersed deformable objects (particles, fibres, beams, 2D/3D deformable solids) with the surrounding viscous fluid and provides a direct link to micro-scale and multi-physics phenomena such as thermal fluctuation, osmosis effects, and various electromagnetic forces. The developed models and methods will assist in understanding biological systems, motivating a new generation of research ideas for computational biomechanics, in particular the formulation of new synthetic materials mimicing nature as well as further development of various micromanipulation techniques such as microneedle, micropipet, poker, and optical tweezer.

The leading challenges in science and technology of this century are clearly the quantitative understanding of biosystems. The research focus is multi-scale and multi-physics modeling of biosystems based on immersed boundary/continuum methods. Human blood is a biological fluid composed of deformable cells, proteins, platelets, and plasma. Human circulatory systems have evolved to supply nutrients and oxygen to and carry the waste from the cells of multicellular organisms through the transport of blood. In the study of the heart, arteries, veins, microcirculation, and pulmonary blood flow, multi-scale and multi-physics coupling of fluids and solids plays an important role. In simulation models, many blood constituents should be represented as immersed flexible shells/beams and solids with various material properties. Furthermore, such models can also be extended to various cardiovascular implants. The closing dynamics of the mechanical valves creates pressure transients that excessively load cells, valve structures, and surrounding tissues, and form cavitation bubbles, which on collapse can cause hemolysis and thrombus initiation. In reality, large motions of flexible structures immersed in biological fluids not only contribute to complex macroscopic stagnation and regurgitation flow behaviors but also affect microscopic chemical/physical changes due to their interaction with proteins, cells, and particles. In fact, the major problems of existing cardiovascular implants can be traced back to the lack of effective modeling tools. These tools are essential for the fine tuning of the designs according to individual organ sizes and physiological flow conditions as well as better understanding of fatigue lifespan of biocompatible materials and atherosclerosis/thrombotic processes. Currently, the intricate structural behaviors, in particular those of immersed flexible shells/beams and solids, are still not well understood. This is due to the enormous difficulties in combining complex nonlinear structural motions with equally complex fluid motions. The goal of my research work is to overcome these difficulties by developing new immersed boundary/continuum methods which will provide a platform for effective modeling of highly deformable shells/beams and solids immersed in biological fluids and facilitate further research in multi-scale and multi-physics coupling of complex fluid-solid systems with microscopic models.

With the increase of machine speeds in paper, thin film, and textile manufacturing processes, the vibration and instability of the traveling web and the web release from roll surfaces have received much attention. To couple with the experimental investigation, a reliable numerical procedure is often needed in modeling the static and dynamic behaviors of three-dimensional moving sheets. It has been found that Kirchoff-theory-based finite elements produce inaccurate or incorrect results even if they satisfy C¹ continuity. In the current research, a mixed finite element formulation based on Mindlin/Reissner plate theory is developed for a moving orthotropic paper sheet. The finite element interpolations are selected according to the MITC plate bending elements, which have been recently proved numerically to satisfy the inf-sup condition. The development of this reliable formulation for orthotropic moving sheets is the key to further investigations of edge flutter and creping process.

Displacement, rotation, and stress waves propagate in a moving thin material with or without tension.

Moving Orthotropic Thin Plates Simulation (Movies)

(z-displacement) (x-rotation) (y-rotation) (stress-xx) (stress-yy) (stress-xy) (stress-xz) (stress-yz)

Instability Analysis of Fluid-Solid Systems and Control of Chaos