Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: The scalar case

We develop a one--parameter family of hp-version discontinuous Galerkin finite element methods for the numerical solution of quasilinear elliptic equations in divergence-form in a bounded Lipschitz domain. Using Brouwer's Fixed Point Theorem, we show existence and uniqueness of the solution. In addition, we derive an error bound in a broken energy norm which is optimal in h and mildly suboptimal in p

A planetary system with an escaping Mars

The chaotic behaviour of the motion of the planets in our Solar System is well established. In this work to model a hypothetical extrasolar planetary system our Solar System was modified in such a way that we replaced the Earth by a more massive planet and let the other planets and all the orbital elements unchanged. The major result of former numerical experiments with a modified Solar System was the appearance of a chaotic window at kappa(E) is an element of (4, 6), where the dynamical state of the system was highly chaotic and even the body with the smallest mass escaped in some cases. On the contrary for very large values of the mass of the Earth, even greater than that of Jupiter regular dynamical behaviour was observed. In this paper the investigations are extended to the complete Solar System and showed, that this chaotic window does still exist. Tests in different 'Solar Systems' clarified that including only Jupiter and Saturn with their actual masses together with a more 'massive' Earth (4 < kappa is an element of < 6) perturbs the orbit of Mars so that it can even be ejected from the system. Using the results of the Laplace-Lagrange secular theory we found secular resonances acting between the motions of the nodes of Mars, Jupiter and Saturn. These secular resonances give rise to strong chaos, which is the cause of the appearance of the instability window. (c) 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinhei

The stability of the terrestrial planets with a more massive 'Earth'

Although the long-term numerical integrations of planetary orbits indicate that our planetary system is dynamically stable at least +/- 4 Gyr, the dynamics of our Solar system includes both chaotic and stable motions: the large planets exhibit remarkable stability on gigayear time-scales, while the subsystem of the terrestrial planets is weakly chaotic with a maximum Lyapunov exponent reaching the value of 1/5 Myr(-1). In this paper the dynamics of the Sun-Venus-Earth-Mars-Jupiter-Saturn model is studied, where the mass of Earth was magnified via a mass factor kappa(E). The resulting systems dominated by a massive Earth may serve also as models for exoplanetary systems that are similar to ours. This work is a continuation of our previous study, where the same model was used and the masses of the inner planets were uniformly magnified. That model was found to be substantially stable against the mass growth. Our simulations were undertaken for more than 100 different values of kappa(E) for a time of 20 Myr, and in some cases for 100 Myr. A major result was the appearance of an instability window at kappa(E)approximate to 5, where Mars escaped. This new result has important implications for theories of the planetary system formation process and mechanism. It is shown that with increasing kappa(E) the system splits into two, well-separated subsystems: one consists of the inner planets, and the other consists of the outer planets. According to the results, the model becomes more stable as kappa(E) increases and only when kappa(E) >= 540 does Mars escape, on a Myr time-scale. We found an interesting protection mechanism for Venus. These results give insights also into the stability of the habitable zone of exoplanetary systems, which harbour planets with relatively small eccentricities and inclinations

Chaos control with ion propulsion

The escape dynamics around the triangular Lagrangian point L-5 in the real Sun-Earth-Moon-Spacecraft system is investigated. Appearance of the finite-time chaotic behavior suggests that widely used methods and concepts of dynamical system theory can be useful in constructing a desired mission design. Existing chaos control methods are modified in such a way that we are able to protect a test particle from escape. We introduce initial condition maps (ICMs) in order to have a suitable numerical method to describe the motion in high-dimensional phase space. Results show that the structure of ICMs can be split into two well-defined domains. One of these two parts has a regular contiguous shape and is responsible for long-time escape; it is a long-lived island. The other one shows a filamentary fractal structure in the ICMs. The short-time escape is governed by this object. This study focuses on a low-cost method that successfully transfers a reference trajectory between these two regions using an appropriate continuous control force. A comparison of the Earth-Moon transfer is also presented to show the efficiency of our method

Stiff oscillatory systems, delta jumps and white noise

Two model problems for stiff oscillatory systems are introduced. Both comprise a linear superposition of N >> 1 harmonic oscillators used as a forcing term for a scalar ODE. In the first case the initial conditions are chosen so that the forcing term approximates a delta function as N tends to infinity, and in the second case so that it approximates white noise. In both cases the fastest natural frequency of the oscillators is O(N). The model problems are integrated numerically in the stiff regime where the time step is of size O(1/N). The convergence of the algorithms is studied in this case in the limit of N tending to infinity and the time step tending to zero. For the white noise problem both strong and weak convergence are considered

Pengaruh Aspek Rasio (Hw/lw) Terhadap Pola Retak Dan Momen Kapasitas Pada Dinding Geser Bertulangan Horiontal Dengan Kekangan Di Bawah Pembebanan Siklik (Quasi-statis)

Dinding geser salah satu elemen struktur yang kaku yang dapat menahan beban lateral dan dapat digunakan sebagai salah satu elemen penting pada bangunan bertingkat. Perencanaan dinding geser serupa dengan kolom namun berbeda pada tulangan horizontalnya. Tulangan horizontal pada kolom dapat sekaligus berfungsi sebagai sengkang, berbeda pada dinding geser. Penelitian ini merupakan penelitian lanjutan dari penelitian sebelumnya. Benda uji yang digunakan pada penelitian ini merupakan benda uji yang awalnya beraspek rasio 2 yang kemudian dipotong menjadi aspek rasio 1,5. Pengaruh aspek rasio pada pola retak, DGK-150-1,5 memiliki jarak antar retak yang lebih renggang dan lebih menyebar di bagian dinding geser dibanding DGK-150-2. Ditinjau dari momen kapasitas, DGK-150-1,5 memiliki nilai yang yang hampir mendekati antara keduanya. Untuk benda uji beraspek rasio sama, pola retak pada DGK-150-1,5 menghasilkan retak yang lebih panjang dikarenakan adanya kekangan dan didominasi oleh retak baru ataupun pertambahan panjang retak. Sedangkan SW-50-1,5 retak yang terjadi tidak sepanjang DGK-150-1,5 dan didominasi oleh petambahan retak dan penyambungan antar retak. Berdasarkan momen kapasitas, DGK-150-1,5 dan SW-50-1,5 ditinjau dengan jarak yang sama dan mutu beton yang berbeda, dimana DGK-150-1,5 menghasilkan momen kapasitas yang lebih besar dibanding dengan SW-50-1,5

Strong and auxiliary forms of the semi-Lagrangian method for incompressible flows

We present a review of the semi-Lagrangian method for advection-diusion and incompressible Navier-Stokes equations discretized with high-order methods. In particular, we compare the strong form where the departure points are computed directly via backwards integration with the auxiliary form where an auxiliary advection equation is solved instead; the latter is also referred to as Operator Integration Factor Splitting (OIFS) scheme. For intermediate size of time steps the auxiliary form is preferrable but for large time steps only the strong form is stable

Presynaptic partner selection during retinal circuit reassembly varies with timing of neuronal regeneration in vivo

Whether neurons can restore their original connectivity patterns during circuit repair is unclear. Taking advantage of the regenerative capacity of zebrafish retina, we show here the remarkable specificity by which surviving neurons reassemble their connectivity upon regeneration of their major input. H3 horizontal cells (HCs) normally avoid red and green cones, and prefer ultraviolet over blue cones. Upon ablation of the major (ultraviolet) input, H3 HCs do not immediately increase connectivity with other cone types. Instead, H3 dendrites retract and re-extend to contact new ultraviolet cones. But, if regeneration is delayed or absent, blue-cone synaptogenesis increases and ectopic synapses are made with red and green cones. Thus, cues directing synapse specificity can be maintained following input loss, but only within a limited time period. Further, we postulate that signals from the major input that shape the H3 HC's wiring pattern during development persist to restrict miswiring after damage

Sensitivity analysis of leaching process on rare earth elements by using metsim software

Leaching is part of the hydrometallurgical treatment in the separation of rare earth elements (REEs). The increase of demand for REEs in the world but its limited supply caused by the separation process that has negative environmental impact, as well as high costs of laboratory work, the alternative method to study on the extraction of these resources are urgently needed. This can be done through simulation study that eliminates the number of experiment that needs to be carried out. In this work, a sensitivity analysis of the leaching process for light rare earth elements (LREEs); Lanthanum (La) and Neodymium (Nd) from monazite concentrate was carried out by employing a software called METSIM which is able to model metallurgical processes. METSIM software is also able to calculate mass and heat balance of complicated hydrometallurgy processes and furthermore, its function has been expanded to involved chemical reactions, process control and equipment sizing. The simulation of the digestion and the leaching process was run and compared with experimental work from the literature, which aimed to optimize the leaching process. Leaching is an extraction process of a substance from a solid material that is dissolved in a liquid. In this study, the leaching process is modelled as a two-steps process, in which the first step is the digestion process, followed by the actual leaching process. The monazite concentrate is made strong bonds as REEs oxide, therefore, they are not soluble in water before leaching process. For the digestion process, the monazite concentrate was mixed with sulfuric acid (H2SO4) in the digestion reactor. The precipitates which contain leachate was then mixed with deionised water in the leaching reactor to increase the solubility of La and Nd. The reaction equations for the digestion reactor are as shown in Equations (1) and (2), while for the leaching reactor are shown by Equations (3) and (4) as follows