\nabla \cdot \left( \mu_r^ \nabla_t^2 E_z= 0 We will additionally assume that our modeling domain is purely vacuum, so that the frequency domain Maxwell’s equations reduce to: Let’s limit ourselves in this blog post to considering only 2D problems, where the electromagnetic wave is propagating in the x-y plane, with the electric field polarized in the z-direction. (For a description of the key differences between these modules, please see my previous blog post, titled “ Computational Electromagnetics Modeling, Which Module to Use?“) These modules provide similar interfaces for solving the frequency domain form of Maxwell’s equations via the finite element method. Such models can be built using the Electromagnetic Waves, Frequency Domain formulation in the RF Module or the Wave Optics Module. We only want to model a small region around the antenna. We may be building this model to simulate an antenna on a satellite in deep space or, more often, an antenna mounted in an anechoic test chamber.Īn antenna in infinite free space. We are often interested in modeling a radiating object, such as an antenna, in free space. Today, we will look at using scattering boundary conditions and perfectly matched layers for truncating domains and discuss their relative merits. COMSOL Multiphysics offers several solutions for this. Thinking about the problem, assume that we arrived at the followingĪpproximations (make sure you understand how we arrived at followingĪpproximations for your future quiz and test): The temperature of the heater isĬonstant at 400K.When solving wave electromagnetics problems, it is likely that you will want to model a domain with open boundaries - that is, a boundary of the computational domain through which an electromagnetic wave will pass without any reflection. We wish to determine the temperature distribution within the sheath. The entireĪssembly is immersed in a fluid and the system is at steady-state, as shown below. Thickness 0.05 m and which starts 0.05 m away from the center. The order isĪlso variable depending on the complexity of the model.Ĭonsider a cylindrical heating rod which is sheathed by a concentric tube of Not all of these steps are always necessary when building a model. Display the desired results in the most meaningful way (Results). Adjust solver parameters and compute (Study).ġ0. Choose the element size to be used (Mesh).Ĩ. you will need to enter these for Laminar Flow and again for Heatħ. You are using (This will be entered separately for each different physics you are Select the boundary, bulk and initial conditions for your system for each physics Select the materials you wish to use in your model (Materials).Ħ. Define the geometry of the model (Geometry).ĥ. Define the parameters, equations and variables pertinent to the model (subĤ. Of study you wish to perform (Time dependant or stationary).ģ. Work through the COMSOL Model Wizard which will require you to select theĬoordinate system for the model, the relevant physics to the problem, and the type The packages are cross-platform (Windows, Mac, Linux,Unix.) InĪddition to conventional physics-based user-interfaces, COMSOL MultiphysicsĪlso allows for entering coupled systems of partial differential equations (PDEs).Ģ. Toolboxes for a large variety of programming, preprocessing and postprocessing Select the materials you wish to use in your model (Materials).ĬOMSOL Multiphysics (formerly FEMLAB) is a finite element analysis, solverĪnd Simulation software / FEA Software package for various physics andĮngineering applications, especially coupled phenomena, or multiphysics.ĬOMSOL Multiphysics also offers an extensive interface to MATLAB and its Define the geometry of the model (Geometry). Define the parameters, equations and variables pertinent to the model (sub directory (Global Definitions). Work through the COMSOL Model Wizard which will require you to select the coordinate system for the model, the relevant physics to the problem, and the type of study you wish to perform (Time dependant or stationary). The packages are cross-platform (Windows, Mac, Linux,Unix.) In addition to conventional physics-based user-interfaces, COMSOL Multiphysics also allows for entering coupled systems of partial differential equations (PDEs). COMSOL Multiphysics also offers an extensive interface to MATLAB and its toolboxes for a large variety of programming, preprocessing and postprocessing possibilities. COMSOL 4.2 Tutorial COMSOL Multiphysics (formerly FEMLAB ) is a finite element analysis, solver and Simulation software / FEA Software package for various physics and engineering applications, especially coupled phenomena, or multiphysics.
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This increase has been documented by the Intergovernmental Panel on Climate Change and NOAA, and thus it is important to note that NOAA's tidal harmonic predictor doesn't consider that the sea level is rising due to steric expansion of warming oceans, increases in overall ocean volume due to displacement through glacial ice melt, and weather-driven fluctuations in local sea level. in the past 100 years, and extrapolation of this trend suggests that it will continue to cause sea levels to rise by this much again (at a minimum) by the year 2100! This represents an increase in eustatic sea level of 1.53 ft. Thus, the 2100 king tide harmonic prediction doesn't seem particularly dramatic because this estimate doesn't account for the 4.66mm/yr average annual increase in sea level NOAA has documented at Sewells Point over the last 90+ years. This prediction method deconstructs the gravitational influences on Earth's oceans by the sun and moon with the assumption that none of these object's orbits are altered. For example, check out this king tide tidal harmonic prediction for Sewells Point, VA, in the year 2100. NOAA's tide prediction charts use harmonic analysis prediction algorithms to estimate impacts of tides months, decades, or even centuries in advance. Select a site on NOAA's map near you to select a site to help map the king tide: Tide predictions for subordinate sites are generated by referencing high/low tide predictions for a nearby designated harmonic site, and then making time and height adjustments to correct the high/low predictions at the subordinate site. Subordinate tidal predictions do not have tidal harmonic constants available.Tide predictions for Harmonic sites are generated directly from the Harmonic constants and have higher confidence than nearby "Subordinate" sites. 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Finally T represents the matter, the energy-momentum, existing in space-time. Note that M by itself does not have such geometric structure there are no distances between points in M alone, no straight lines, and so forth. The metric tensor defines the metric and geometric structure of the space-time: distances between points A and B, whether points A, B and C are collinear, whether line L is a straight line (geodesic) or curved, and so on. For example some models of GTR have M structurally identical to □ 4, which means that space-time can be coordinatized (all the points labeled) with four-dimensional Cartesian coordinates. The manifold is a collection of points with a local and global topology built-in. ) Notice that each of these objects is four -dimensional, representing not just how things are at a specific time but rather how things are over the entire history of the (model-) universe. (In the latter case we say the space-time is "empty," but it may still have an interesting structure as encoded in g. Like g, T is defined everywhere in the space-time, but unlike g, T may be exactly equal to zero at some or even all points of space-time. Einstein's field equations describe these interactions, and delimit the set of models, or physically possible worlds, corresponding to the theory.Ī model of GTR is usually presented as a triple consisting of a four-dimensional, continuously differentiable manifold M, a metric-field tensor g (representing the geometry of space-time) defined everywhere on the manifold, and a stress-energy tensor T representing the material substances in space-time. GTR describes the dynamical interaction of material substances in space-time with other material substances, as well as their interactions with the variably-curved structure of space-time itself. Subsequently philosophers have explored the status of general covariance, and therefore of the hole argument, in the domain of quantum gravity theories. Regardless of which viewpoint is better supported, it is indisputable that Earman and Norton's hole argument led to a huge resurgence of interest in the interpretation of space-time in GTR, and lies at the core of much of the philosophy of space-time theories published since 1987. Instead most philosophers came to think that the hole argument's indeterminism is merely an artifact of a particular interpretation of the mathematical structure of GTR that we are not logically compelled to accept. But within a few years this view of the argument's significance was widely rejected. Earman and Norton argued that the problem is reason enough to justify rejecting a substantival view of space-time in GTR. A close cousin of Einstein's hole argument was put forth by John Earman and John Norton (1987) as an argument claiming to show that, if one embraces a substantival view of space-time, then in a generally covariant theory such as the GTR, one is committed to an unpleasant form of indeterminism. Seven decades later, after the rediscovery of Einstein's argument by John Stachel and John Norton, history repeated itself. The indeterminism allegedly shown by the hole argument is spurious, and the argument cuts no ice in favor of any particular theory or interpretation of the nature of space-time. From his second point of view the argument rests on a mistaken interpretation of the mathematics of general covariance. The second use of the hole argument came in 1915 when Einstein came to see the argument, taken in its first form, as a mistake. Einstein was not fully satisfied with that theory, in part because he believed that general covariance was necessary if a theory were to capture a fully general relativity of motion, and so the hole argument served to help Einstein reconcile himself (temporarily and only partially) to the Entwurf theory. First before the discovery of his final field equations for the General Theory of Relativity (GTR), the argument was put forward as a justification for accepting non -generally covariant field equations, namely those of the 1913 Einstein-Grossman Entwurf theory. Einstein put the argument to two different uses. The point of the argument may be put as follows: If a physical theory's equations are generally covariant (that is, invariant under a wide group of continuous coordinate transformations) then the theory is in a certain specific sense indeterministic. The original "hole argument" ( lochbetrachtung ) was created by Albert Einstein. |
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