a b s t r a c tThe hydrodynamic force exerted by a fluid on small isolated rigid spherical particles are usually well described by the Maxey-Riley (MR) equation. The most time-consuming contribution in the MR equation is the Basset history force which is a well-known problem for many-particle simulations in turbulence. In this paper a novel numerical approach is proposed for the computation of the Basset history force based on the use of exponential functions to approximate the tail of the Basset force kernel. Typically, this approach not only decreases the cpu time and memory requirements for the Basset force computation by more than an order of magnitude, but also increases the accuracy by an order of magnitude. The method has a temporal accuracy of OðDt 2 Þ which is a substantial improvement compared to methods available in the literature. Furthermore, the method is partially implicit in order to increase stability of the computation. Traditional methods for the calculation of the Basset history force can influence statistical properties of the particles in isotropic turbulence, which is due to the error made by approximating the Basset force and the limited number of particles that can be tracked with classical methods. The new method turns out to provide more reliable statistical data.
In this article, we present a least-squares method to compute freeform surfaces of a lens with parallel incoming and outgoing light rays, which is a transport problem corresponding to a non-quadratic cost function. The lens can transfer a given emittance of the source into a desired illuminance at the target. The freeform lens design problem can be formulated as a Monge-Ampère type differential equation with transport boundary condition, expressing conservation of energy combined with the law of refraction. Our least-squares algorithm is capable to handle a non-quadratic cost function, and provides two solutions corresponding to either convex or concave lens surfaces.
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