An efficient threshold dynamics method for topology optimization for fluids

We propose an efficient threshold dynamics method for topology optimization for fluids modeled with the Stokes equation. The proposed algorithm is based on minimization of an objective energy function that consists of the dissipation power in the fluid and the perimeter approximated by nonlocal energy, subject to a fluid volume constraint and the incompressibility condition. We show that the minimization problem can be solved with an iterative scheme in which the Stokes equation is approximated by a Brinkman equation. The indicator functions of the fluid-solid regions are then updated according to simple convolutions followed by a thresholding step. We demonstrate mathematically that the iterative algorithm has the total energy decaying property. The proposed algorithm is simple and easy to implement. A simple adaptive time strategy is also used to accelerate the convergence of the iteration. Extensive numerical experiments in both two and three dimensions show that the proposed iteration algorithm converges in much fewer iterations and is more efficient than many existing methods. In addition, the numerical results show that the algorithm is very robust and insensitive to the initial guess and the parameters in the model.Comment: 23 pages, 24 figure

Statistical exponential formulas for homogeneous diffusion

Let $\Delta^{1}_{p}$ denote the $1$-homogeneous $p$-Laplacian, for $1 \leq p \leq \infty$. This paper proves that the unique bounded, continuous viscosity solution $u$ of the Cauchy problem \left\{ \begin{array}{c} u_{t} \ - \ ( \frac{p}{ \, N + p - 2 \, } ) \, \Delta^{1}_{p} u ~ = ~ 0 \quad \mbox{for} \quad x \in \mathbb{R}^{N}, \quad t > 0 \\ \\ u(\cdot,0) ~ = ~ u_{0} \in BUC( \mathbb{R}^{N} ) \end{array} \right. is given by the exponential formula $$u(t) ~ := ~ \lim_{n \to \infty}{ \left( M^{t/n}_{p} \right)^{n} u_{0} } \,$$ where the statistical operator $M^{h}_{p} \colon BUC( \mathbb{R}^{N} ) \to BUC( \mathbb{R}^{N} )$ is defined by $$\left(M^{h}_{p} \varphi \right)(x) := (1-q) \operatorname{median}_{\partial B(x,\sqrt{2h})}{ \left\{ \, \varphi \, \right\} } + q \operatorname{mean}_{\partial B(x,\sqrt{2h})}{ \left\{ \, \varphi \, \right\} } \,$$ with $q := \frac{ N ( p - 1 ) }{ N + p - 2 }$, when $1 \leq p \leq 2$ and by $$\left(M^{h}_{p} \varphi \right)(x) := ( 1 - q ) \operatorname{midrange}_{\partial B(x,\sqrt{2h})}{ \left\{ \, \varphi \, \right\} } + q \operatorname{mean}_{\partial B(x,\sqrt{2h})}{ \left\{ \, \varphi \, \right\} } \,$$ with $q = \frac{ N }{ N + p - 2 }$, when $p \geq 2$. Possible extensions to problems with Dirichlet boundary conditions and to homogeneous diffusion on metric measure spaces are mentioned briefly

Asymptotic statistical characterizations of p-harmonic functions of two variables

Generalizing the well-known mean-value property of harmonic functions, we prove that a p-harmonic function of two variables satisfies, in a viscosity sense, two asymptotic formulas involving its local statistics. Moreover, we show that these asymptotic formulas characterize p-harmonic functions when 1 < p < \infty. An example demonstrates that, in general, these formulas do not hold in a non-asymptotic sense.Comment: Minor changes from published version: updated author info., one updated referenc

Modeling Ultrasonic field Emanating from Scanning Acoustic Microscope for Reliable Characterization of Pathogens (Biological Materials)

Acoustic microscopy provides extraordinary advantages over state-of-the-art invasive imaging techniques to determine the mechanical properties of living colonies of pathogens and micro-organisms. It is possible to obtain the morphomechanical parameters of the pathogenic colonies e.g. variation of thickness, stiffness and the coefficients of attenuation, using scanning acoustic microscope (SAM). However, the process requires an expert with extensive understanding of SAM and ultrasonic signals which is very time consuming and expensive for complex form of analysis. Due to lack of a suitable computational tool, presently the ultrasonic wave scattering, reflection and transmission through the biological specimens cannot be properly visualized. Without any reliable simulated environment, it is extremely difficult to extract the morphomechanical parameters from the invading pathogens. To understand the ultrasonic signals that are reflected or scattered back from the biological specimens, one would need to compute the Pupil Function (PF), i.e. generated by a particular SAM lens. PF is the total pressure field in front of the lens at focal plane generated by the lens and cannot be experimentally measured without placing a reflecting surface in front of the lens. Hence to determine the PF one could change the interpretation of PF. Thus a detailed computer simulation platform for the SAM experiments is necessary. Particularly it is mandatory to obtain the accurate PF that is generated by a particular SAM lens used in the experiments before decoding the morphomechanical properties of the biological specimens. To obtain the accurate PF in front of an acoustic lens, this dissertation presents a detailed development of Distributed Point Source Method (DPSM) for modeling SAM experiments. The ultrasonic field in front of the focused 100 MHz lens, obtained from the simulation can be further used to determine the material properties of the biological specimens. An accurate modelling of SAM lens using the distributed point source method (DPSM) is proposed for its proven capability to simulate ultrasonic fields at higher frequencies. DPSM is computationally cheap and efficient than the Finite Element Method (FEM). The acoustic lenses used in the SAM are commonly made of sapphire but enclosed with a brass casing. The sapphire head consists of four different geometrical shapes and each segment has individual influences on the visualization of the ultrasonic field produced by the transducer. Thus, the accurate geometry of the acoustic lens is an important factor for modeling. Using the DPSM accurate geometry of a 100 MHz lens is modeled and the PF is computed in front of the lens. It is shown that as per the design specification of the lens, the pressure field is accurately focused at the focal point. The peak pressure at the focal point and the rippled wave effect away from the focal point are verified in the DPSM based simulation environment

A New Multiscale Representation for Shapes and Its Application to Blood Vessel Recovery

In this paper, we will first introduce a novel multiscale representation (MSR) for shapes. Based on the MSR, we will then design a surface inpainting algorithm to recover 3D geometry of blood vessels. Because of the nature of irregular morphology in vessels and organs, both phantom and real inpainting scenarios were tested using our new algorithm. Successful vessel recoveries are demonstrated with numerical estimation of the degree of arteriosclerosis and vessel occlusion.Comment: 12 pages, 3 figure