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• However, the techniques used listed below are applicable in a much wider context.
• The Grid dataset adds volumetric grid cells in order that boundary element quantities can be visualized and evaluated in void domains.
• the Earth–Moon system in the Graphics window.
• how we can use the Coefficient Form PDEinterface to define a PDE simply by matching the coefficients.

[newline]The surface plots shows the gravity on the surfaces, the contours show the gravity strength, and the arrows show the direction of the gravitational field.
Which is used to assign fixed values of the dependent variable on the boundary.
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The model used throughout this short article can be acquired for download.
You can open and explore the model files to see exactly how the Laplace and Poisson’s equations for the gravitational field of the Earth–Moon system were solved.
In anotherLearning Center article we will examine the Coefficient Form PDE more.
By default, domain 1 and 2 and the Infinite void domain are selected for the PDE, Boundary Elements.

The Grid dataset adds volumetric grid cells so that boundary element quantities could be visualized and evaluated in void domains.
To make computations easier and also formulate a scalar-valued PDE, we will also need the gravitational potential, which is a scalar field, , with the somewhat unfamiliar unit of .
We will have later the best way to utilize the Mathematics interfaces to specify custom units and integrate them with the automatic unit handling capabilities in the program.
In this visualization of the Earth–Moon system, the slice plot shows the gravity in space for values below 0.05 m/s2.

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Note that we need not enclose both spheres in a volume mesh due to the fact that we will use a boundary element method for the gravity in space.
This model could alternatively be thought as a 2D axisymmetric model.
Since we will model the Earth–Moon system being an idealized and isolated system, we do not desire to impose unnecessary and artificial boundary conditions.

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• In subsequent parts, we will read more about the mathematics interfaces and how to model more general PDEs and systems of PDEs.
• Since we shall model the Earth–Moon system as an idealized and isolated system, we usually do not want to impose unnecessary and artificial boundary conditions.
• For electrostatics, corrosion, magnetostatics, and acoustics, you can find dedicated physics interfaces

BEM runs on the mesh on surfaces only, i.e., not in the computational volume.
This is advantageous once the computational volume is large, which typically happens once the objects of interest are far apart.
The Earth and the Moon are relatively far apart compared to their radii, making modeling the area that surrounds them suitable for BEM.

Where G and rho are defined as a parameter and a variable for the gravitational constant and density elsewhere in the model, respectively.
The value of rho will be that of the Earth, 5515; and the Moon, 3340, within both spheres that we will use to represent these bodies.
The gravity equation modeled utilizing the Coefficient Form PDEinterface.
The settings for the plot where in fact the built-in variable is used in the expression field.

Utilizing The Coefficient Form Pde For Poisson’s Equation

The table displaying the surface gravity, generated after clicking on the surface of every sphere in the results plot.
ASurface plot displaying the magnitude of surface gravity on both planetary bodies.
The expression found in a surface plot to visualize the magnitude of the top gravity.

It is possible to combine equations of different types to form an array of nonlinear equation systems, modeling a myriad of mathematical and physical processes.
Most of these interfaces, with the exception of some of those beneath the Optimization and Sensitivity and Moving Interface branches, are available as part of the core functionality in the software.
With respect to the finite element mesh found in the spheres, a fixed element size of 500 km is used for the Earth, as shown in the figure below, and 200 km can be used for the Moon.
This is accomplished by setting exactly the same value for the utmost and minimum element size.
Inside of the spheres there will be tetrahedral finite elements.

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Using a mouse, we are able to click on the surface of each sphere and read off the values 9.81for the Earth and 1.62for the Moon, which matches the expected values within significantly less than a percent.
This concludes the setup of the finite-element-based Coefficient Form PDE. In the next step we add a PDE, Boundary Elements interface.
The geometry simply includes two spheres at the distance d_moonapart, as shown in the figure below.
In the example of Newtonian gravity, we will now see how we can use the Coefficient Form PDEinterface to define a PDE simply by matching the coefficients.