Modeling Electromagnetics

Hi to everyone and good morning. My name is Sebastian Garcia. And well, this is a first talk of a series of how to model electromagnetics with the finite element method in Mathematica. And first of all this is going to be focused on electrostatics. So what we are going to address today. Well, first of all I want to show you that now we have a complete workflow for modeling electrostatic models. Then what I want to show you is how to model multimaterials in the electromagnetics fields. And then I want to show you how you can use these new functionalities to make multiphysics models. And in this case we are going to see how to model multiphysics actuator. In this case we are going to couple an electrostatic model with a solid mechanics model. And for example, in this image you can already see that you can compute many things. In this case this is a 2D cross sectional area of a capacitor where you can add different materials, for example a dielectric and a vacuum that it is running it. So these are the things, the main things that we are going to address today. So basically what we are doing in in this subfield of electromagnetism, that is electrostatics, we are going to simulate the electrostatic phenomenal that it is electric fields that are produced by stationary charges. And these stationary charges can be introduced by a potential difference or surface charge densities, etc. And these new functionalities are presented with these new functions. The these are the PD components and boundary conditions that we have introduced for this subfield of electromagnetism. This is the first one that generates the equation that we need for the electrostatic equation. And then we have the boundary conditions. Essentially it is the electric potential condition that it is a difficult condition. And then we have our Norman values that we can use in our models. So first of all we have the electrostatic PD component. This new function will generate automatically your electrostatic equation. And the capabilities of this function is that you can add your constitutive relation. For example, you can add a polarization vector or a relative permittivity of both at the same time. If you want to describe a material that it is better described with a polarization or with a relative permittivity. We can deal with non linearities such as a remanent polarization. And then we have other kind of parameters like adding sources and that goes with volume charge densities. Or we can add extra parameters like thickness or cross sectional area. When we want to apply symmetries in our model and scale the equation so Basically, this is a way one can generate the equation giving the parameters that you want. So the most important thing is that we already have an electrostatic monograph where the whole workflow is a scribe. First of all, we have an overview example where we show a 3D capacitor. In this 3D capacitor we show how to compute the electric fields generated. And then how to compute the electric fields, the electric flux density and then different methods to compute capacitance value with different methods. Then we have our electrostatic equation section which where we give the whole theory behind how to model electrostatic. And then we give many different examples that showcase the different variations of the electrostatic equation. These include anisotropic relative permittivities or multiple materials and non linearities. Then we have a third section where we can add point charges, we can add line charges, densities inside our domain for whatever reason you want. Then we have a section that it is named secondary quantities. In these secondary quantities, our focus was to show you how to make post processing after doing an electrostatic model. In this case, we wanted to show you how to compute for example, capacitance with the different methods that they are available. Then for example, how to compute effectively the fields, the electric fields and also how to compute forces. This is really useful because you can compute electrostatic forces between electrodes and and between dielectrics. See how much is the force between these two objects. And we introduce how to compute these forces with a really general method that it is the Maxwell stress tensor. So it is really helpful to compute this kind of quantities. And then we have the boundary conditions we where we show you how to apply properly the boundary conditions for electrostatic models. And this has different conditions and also how to use periodic boundary conditions for electrostatic models. So all this information is in this monograph that you can access to the new documentation of 14 oh version of Mathematica. So first of all I want to show you how we are dealing with multiple materials in the electromagnetics context and in this case in the electrostatics. Because there is a really big difference between modeling, for example in heat transfer than in electrostatic. And the main reason is that in electromagnetic magnetism it is sometimes most useful to know the electric fields, the D or the E field. Meanwhile, in the heat transfer there is no reason for compute derivatives, our main variable. And in this case it is necessary for. So for example, in this case one can model a 3D capacitor. In this case you model the dielectric region between the electrodes. And also you can Model the the volume surrounding the capacitor that it can be vacuum air, or it's in this case a 2D model of the same capacitor. So what happens at the interface when fields are computed from electrostatic model with multiple materials, One can see discontinu and these discontinuities arise because we have different materials. So these in some cases can have issues. When doing multiple materials and computing fields, you can see the way that these fields are computed are with these formulas. You get the gradient of the electric potential and then you get the E field. And if you want to get the the electric flux density, then you multiply it by the relative permittivity and the vacuum permittivity. But as I told you, what if you have the case that you have different relative permittivities and it is inhomogeneous in your model? For example, in this case, in the 2D capacitor, we have a dielectric permittivity different from the vacuum permittivity that is surrounding the capacitor. So in this case, I'm going to compute the solution for a cross section of a parallel plate capacitor. You can access to this model in the monograph. So we define the variables and the parameters of the equation. If you can see, we are using a piecewise function to define the different values of the relative permittivity. A value of 7 for the dielectric and a value of 1 for the region surrounding the capacitor. Then we define the boundary conditions. They are electric potential conditions. We are applying a voltage difference between the electrodes. Then we compute the electric potential that it returns an interpolating function span over the whole domain. Now that we have this interpolating function, how do we deal with these discontinuities at the interface? The way we did it is by creating a new function that it is discontinuous interpolating function. In this case, I'm going to show you how you can change this value at this node that it is at the interface of our model. So we call the equation and what we have in return is a discontinuous interpolating function that will prioritize one of the two values. In this case, the default value it is 1212 refers to the element markers that we use to specify the different material regions. In the left you can see in orange that it is a dielectric, and in light blue, the vacant region. So in this case we are prioritizing this value with this relative permittivity. So this is the way it is done. And after we compute the discontinuous interpolating function, we can compute the electric field. And this is very easy. You can use the GRAD function and this is the way it is done. And also if you want to compute the D field, you can do it using evaluate on element mesh. This function what gives you the capability to evaluate your relative permittivity that we saw that it is a piecewise function. So this function allows you to compute the D field having different relative permittivities. So here we are also using the default values that will prioritize the dielectric region. After we compute the D field, you can evaluate it at the point that I showed you before. And we have these values and I wanted to show you that if you change the marker priority. So in this case, now we are going to prioritize the value with the relative permittivity from the air instead that from the dielectric. And you can see already that the values are different. So this new workflow for discontinuities at interface gives you more consistency in your model. When you want to compute really specific things like for example, forces, you want to know what is the value inside the dielectric, or you want to know the value outside the dielectric at the same interface. So this is the way that it should be done when computing electric fields. And you have multiple, multiple materials. Then now we can go to the, to our main model that I want to show. It is how to model an electrostatically actuated main device. This is taken from this article inspired from it. And I'm just going to give a quick overview of what is this. Well, a MEM device is a microelectro mechanical device that typically consists of a thin movable beam or electrodes suspended over a fixed electrode. In this case, we are going to model these two electrodes. You can see that our top electrode is movable. It is only fixed at the left. And then we have a bottom electrode that it is all hole fixed. And we apply a voltage difference between these two electrodes. So when applying a voltage difference between two electrodes, we are going to induce surface charge densities over the surface of the electrodes. And these surface charge densities are going to generate an electrostatic force between two electrodes. So as the bottom electrode is fixed, we are going to see only a deformation of the top electrode is going to paint in the downwards direction. And this is what we are going to model by coupling the electrostatic simulation with a solid mechanic simulation. In this case, for the surrounding, we are going to use a zero electric Potential at exterior boundaries. And we are going to specify the voltage with an electric potential condition in the electrodes. You can make more complicated this model by adding, for example, different relative permittivities or making it a 3D model. But in this case, we are going just to simulate it in a 2D domain. So first of all we need to define our mesh. This is really important. The mesh has to be done by removing the electrodes. We always remove the electrodes or the metallic surfaces, or in other words, the conductors from the mesh. So this is the way it is done. Then we define the variables and parameters of our model and then we generate the electrostatic equation. In this case, it is not necessary to define a relative permittivity. Then we are going to use element markers from the boundary conditions. In this case it is more easier to define it the boundary conditions where to place them. So in this case we are going to specify the electric potential conditions in element marker one. This is the exterior boundaries and a zero potential. Also in the element marker two, that that is the bottom electrode. And then we define the electric potential at the top electrode, that is of 20 volts. After solving it, we are going to see this distribution that matches with our boundary conditions. We can see already the voltage difference. And here we can see the gradient of voltages that is going to create these electrostatic force between the electrodes. So the good thing about this model is that you only need to compute once the electrostatic model and then you can go to the solid mechanics models and do the coupling. So in this case, the solid mechanics coupling is going to be with a pressure. So we already solved the electrostatic model. Then we are going to couple both models with a boundary condition. So in solid mechanics, we have boundary conditions that can specify loads pressures. So the pressure that we are going to use to express the deformation is going to be a pressure that it is dependent on the electric flux density of our electrostatic model. After computing the pressure with this electric flux density, we compute the deformation using the solid mechanics workflow. And then we update the electric flux density with this new deformation that we achieve. And then we Repeat steps from 2 to 4 until we get to a convergence that is going to be the maximal deformation and displacement that we are going to see in our model. So in this case, for the solid mechanic model, we just need to model the top electrode. It's not necessary to simulate the whole domain that we use in the electrostatic model. That brings another advantage, and it's very easy to do it in Mathematica. Then we Define our variables and parameters for the solid mechanical analysis that includes the jump modulus, the question ratio thickness. And we are going to use a plane stress formulation in this case. Then we define the boundary conditions. We are fixing the left side of the electrode and then we are specifying the solid boundary load value, where we specify the electrostatic pressure. And the electrostatic pressure function is at the end of our presentation that this already computes the pressure from the electric flux density. If you see, I'm not going to give a displacement yet for this boundary and we are going to compute the first deformation. We define the PD component equation, then we proceed to solve it. So this is going to be our first displacement from the first pressure. And then we are going to do the iteration. In this iteration we are going to use the displacement in the Y directions downwards and maximal displacement. And this is the value that we are going to use if we to test if we have yet to the convergence. So we are going to compute the relative error of the maximal displacement with the all displacement and until we reach an accuracy that we want. So in this case we are going to do the iteration in a while loop with a sundry boundary load value. Here we are going to specify also the electrospatic pressure. And the difference is that now we are giving as an input the displacement that we before computed and then we solve the model. So if we run this loop, we are going to get to a final displacement of our movable electrodes. And at the end we are going to achieve what we want. So in this case, as you can know in the solid mechanics, you can do a lot of post processing, compute stresses, deformations, strains. In this case we computed the bond mises stress, which is the results that are shown in the article and they match with what it is given in the article. So we can see here the distribution of the bond on Mrs. Stress in the deform top electrode. We can see already the distribution and the deformation that we the maximum displacement that we are going to get with these characteristics. So if you can see, the electrostatic model can be coupled already with different physics, for example the solid mechanics. And well, you can do many things already with just electrostatic equation. Then what about our future work in electromagnetism? As I told you, this is one the first of a series of electromagnetics of how to model electromagnetics. So we already working on the workflow for electric currents that follows the same principle as an electrostatic equation. It has the same form, but it applies for conductors and also magnetic fields and electromagnetic field. So we want to show you complete workflows for doing complex models, how to couple them, how to compute secondary quantities. And the main idea that I want to let you see is that you can access to these tutorials that will be very helpful for your future projects about how you can use all these new functionalities. We want to do the same for these fields. I want to emphasize that you can already see simulate these phenomena in electromagnetism, but obviously it requires more work from you to know which equation to use, how to specify boundary conditions, which things we cannot do and what things we can do. So this is the purpose of doing these monographs to show you how. Yeah, how and what you can simulate with Mathematica and the finite element method that we have. So these are the things that we are working on right now. And this was the first addition that we. We did with electrostatics. So this is the end of the presentation and I will be glad to answer you some questions that you have. Let me go to here and stop screen sharing to see if you have questions. Okay. Will this modeling help to improve the transducers technology like microphone or musical instrument pickups or ultrasonic transducers? Yeah, maybe. I haven't seen if there is a coupling between acoustics models and electrostatics, but certainly there should be a way to model these two technologies. I don't know which physics and specific you need to use, but we already have a complete workflow for acoustics and simulate a pressure fields that comes with microphones are ultrasonic. There is the capability of doing these things, but we don't have any examples yet that showcase that. And from other questions, how is electrostatic turbulence handled? I don't know what you mean. With electrostatic turbulence right now the electrostatic equation can simulate different things also non linearities and maybe with more work you can simulate ferroelectrics. Yeah. For 2D cases and axisymmetric cases it is really easy. There is in the electrostatic equation you only need to specify a parameter that it is region symmetry and that goes to axisymmetric is already in the monograph how to simulate axisymmetric models in 2D cases and axisymmetric models in 2D Cases is really easy. We already have a model that shows how to simulate axisymmetric models. It is a spherical capacitor. Then do these simulations can scale up to larger numbers of processors? I don't know that answer, but surely I can send you an email if that is possible or not. I don't know the answer yet, but I Can write you if this can be done to do more complex or more big domains simulations. But in overall, all the capabilities that we have in the different in the different sections of multiphysics, you can see that we you can do the same in electrostatic that in heat transfer in solid mechanics. Just that it makes more work to done. And I see that there is another question. I'm sure in antenna near fear and patterns is these resources that are really available that you're aware of or someone with experience could contact antennas and near firearm patterns they deal with electromagnetic fields. And if it is the case that it is a 3D model, we don't have yet the capabilities to simulate electromagnetic fields propagating in 3D spaces. Because it needs of this famous core vector elements that we don't have yet in Mathematic implemented. So maybe is it you only be possible to simulate antennas and electromagnetic radiation if you do a simplification of your model and that will be to go to a 2D domain instead of a 3D domain. So yeah, basically electromagnetic fields in 3D cases, most of the 3D simulations done in 3D domains will be really difficult to simulate them because you need another kind of elements in our current version of finite element method that we have in Mathematica. So there can be three models that you can simulate. For example, in the magnetic fields, if you only want to simulate time varying magnetic fields, but the coupling between electric and magnetic fields in electromagnetic regime, high frequency regime, it is not possible to do it right now. So yeah, we already have these limitations, but we are working on that to extend it to more to more things in the high frequency regime. Yeah, I think that I have answered all the questions and thank you for attending to this webinar and you can contact me or any other partner that we are in the fem a group of Mathematica. If you want to show you that you want to show us models that you are working on electromagnetics simulations that you already are working on with Mathematica, it will be very helpful for us.