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We propose a new strategy for constructing the p th degree immersed finite element (IFE) spaces by applying the least squares framework in Adjerid et al. (2017) on fictitious elements. This new construction method significantly reduces the ill-conditioning, caused by the small subelement issue in Adjerid et al. (2017), of solving the local IFE shape functions. The proposed IFE spaces are employed in a discontinuous Petrov–Galerkin (DPG) scheme to solve the second order elliptic interface problems. We present a group of numerical examples to show that the DPGIFE method with the new p th degree IFE space as the trial function space has the optimal convergence rate, which improves the numerical results reported in Adjerid et al. (2017).

This article reports our explorations for solving interface problems of the Helmholtz equation by immersed finite elements (IFE) on interface independent meshes. Two IFE methods are investigated: the partially penalized IFE (PPIFE) and discontinuous Galerkin IFE (DGIFE) methods. Optimal convergence rates are observed for these IFE methods once the mesh size is smaller than the optimal mesh size which is mainly dictated by the wave number. Numerical experiments also suggest that higher degree IFE methods are advantageous because of their larger optimal mesh size and higher convergence rates.

A group of partially penalized immersed finite element (PPIFE) methods for second-order hyperbolic interface problems were discussed in Yang (Numer Math Theor Methods Appl 11:272–298, 2018) where the author proved their optimal O(h) convergence in an energy norm under a sub-optimal piecewise $$H^3$$ regularity assumption. In this article, we reanalyze the fully discrete PPIFE method presented in Yang (2018). Utilizing the error bounds given recently in Guo et al. (Int J Numer Anal Model 16(4):575–589, 2019) for elliptic interface problems, we are able to derive optimal a-priori error bounds for this PPIFE method not only in the energy norm but also in $$L^2$$ norm under the standard piecewise $$H^2$$ regularity assumption in the space variable of the exact solution, rather than the excessive piecewise $$H^3$$ regularity. Numerical simulations for standing and travelling waves are presented, which corroboratively confirm the reported error analysis.

This article presents an error analysis of the symmetric linear/bilinear partially penalized immersed finite element (PPIFE) methods for interface problems of Helmholtz equations. Under the assumption that the exact solution possesses a usual piecewise $H^2$ regularity, the optimal error bounds for the PPIFE solutions are derived in an energy norm and the usual $L^2$ norm provided that the mesh size is sufficiently small. A numerical example is conducted to validate the theoretical conclusions., 14 pages, 1 figure

In Lin et al. (2015), a group of typical partially penalized immersed finite element (PPIFE) methods were developed for solving interface problems of the parabolic equations, and its authors proved that these PPIFE methods had the optimal O ( h ) convergence rate in an energy norm under a sub-optimal piecewise H 3 regularity assumption. In this article, we report our reanalysis of these PPIFE methods. Our proofs are based on the optimal error bounds recently derived in Guo et al. (2019) for elliptic interface problems, and we are able to show that these PPIFE methods have the optimal convergence not only in an energy norm but also in the usual L 2 norm with the assumption that the exact solution possesses the standard piecewise H 2 regularity instead of the excessive piecewise H 3 regularity in the space variable. Numerical results are also presented to validate the reported error analysis.

In this paper, a time-fractional heat conduction problem is mathematically proposed for an experimental heat conduction process in a 3-layer composite medium. A numerical solution to the direct problem is obtained with finite difference method. In regard to the inverse problem, the optimal order of Caputo fractional derivative is estimated with Levenberg–Marquardt method. Comparing with the carbon–carbon experimental data, the results show that the time-fractional heat conduction model provides an effective and accurate simulation of the experimental data. The rationality of the proposed time-fractional model and validity of Levenberg–Marquardt method in solving the time-fractional inverse heat conduction problem are also manifested according to the results. By conducting the sensitivity analysis, the feasibility of the parameter estimation is further discussed.

Around 300 bat species are known to emit their ultrasonic biosonar pulses through the nostrils. This nasal emission coincides with the presence of intricately shaped baffle structures surrounding the nostrils. Some prior experimental evidence indicates that these "noseleaves" have an effect on the shape of the animals' radiation patterns. Here, we present a numerical acoustical analysis of the noseleaf of a horseshoe bat species. We show that all three distinctive parts of its noseleaf ("lancet," "sella," "anterior leaf") have an effect on the acoustic near field as well as on the directivity pattern. Furthermore, we show that furrows in one of the parts (the lancet) also exert such an influence. The underlying physical mechanisms suggested by the properties of the estimated near field are cavity resonance, as well as reflection and shadowing of the sound waves emitted by the nostrils. In their effects on the near field, the noseleaf parts showed a tendency toward spatial partitioning with the effects due to each part dominating a certain region. However, interactions between the acoustic effects of the parts were also evident, most notably, a synergism between two frequency-dependent effects (cavity resonance and shadowing) to produce an even stronger frequency selectivity.

Horseshoe bats emit their ultrasonic biosonar pulses through nostrils surrounded by intricately shaped protuberances (noseleaves). While these noseleaves have been hypothesized to affect the sonar beam, their physical function has never been analyzed. Using numerical methods, we show that conspicuous furrows in the noseleaf act as resonance cavities shaping the sonar beam. This demonstrates that (a) animals can use resonances in external, half-open cavities to direct sound emissions, (b) structural detail in the faces of bats can have acoustic effects even if it is not adjacent to the emission sites, and (c) specializations in the biosonar system of horseshoe bats allow for differential processing of subbands of the pulse in the acoustic domain.

Around 300 bat species are known to emit their ultrasonic biosonar pulses through the nostrils. This nasal emission coincides with the presence of intricately shaped baffle structures surrounding the nostrils. Some prior experimental evidence indicates that these “noseleaves” have an effect on the shape of the animals' radiation patterns. Here, a numerical acoustical analysis of the noseleaf of a slit-faced bat species is presented to show that all three distinctive parts of its noseleaf (“pit”, “upper leaf”, “lower leaf”) have an effect on the acoustic near field as well as on the directivity pattern. In their effects on the near field, the noseleaf parts showed a tendency toward spatial partitioning with the effects due to each part dominating a certain region. However, interactions between the acoustic effects of the parts were also evident, most notably, a synergism between the frequency-dependent effects of two parts of the noseleaf (“pit”, “lower leaf”) to produce an even stronger frequency selectiv...

Noseleaves in slit-faced bats have been hypothesized to affect the sonar beam. Using numerical methods, we show that the pit in the noseleaf of an Egyptian slit-faced bat has an effect on focusing the acoustic near field as well as shaping the radiation patterns and hence enhancing the directionality. The underlying physical mechanism suggested by the properties of the effect is that the pit acts as a doubly curved reflector. Thanks to the pit the beam shape is overall directional and more selectively widened at the high end of the biosonar frequency range to improve spatial coverage and detectability of targets.

The nostrils of horseshoe bats are surrounded by elaborate noseleaves which consist of three parts: anterior leaf, sella, and lancet. Of these parts, shape and location of the anterior leaf, which forms a baffle around the nostrils, are most suggestive of an acoustic function. While such an argument is not as readily made for them, the other two parts show more conspicuous structural features than the anterior leaf. In particular, the lancet is furrowed by deep horizontal grooves in most species and flaps can be attached to the bottom of the lancet in some species. The results of a numerical analysis presented here indicate that the lancet furrows and the sella flaps both have effects on the animals’ radiation pattern. The acoustic mechanisms behind these effects differ; however, Lancet furrows act as half‐open resonance cavities whereas the sella flaps have a more complicated impact on the wavefield phase and amplitude. Despite the differences in the acoustic mechanisms, there are two important commonalities in the effects of these structural features: Both act as devices for beam dispersal widening the distribution of sound energy over angle and the effects of both are limited to the FM‐component of the biosonar pulse.