## Integrals with Moving Limits and Solving Integro-Differential Equations

##### Temesgen Kindo October 6, 2016

In a previous blog post, we discussed integration methods in time and space, touching on how to compute antiderivatives using integration coupling operators. Today, we’ll expand on that idea and show you how to analyze spatial integrals over variable limits, whether they are prescribed explicitly or defined implicitly. The technique that we will describe can be helpful for analyzing results as well as for solving integral and integro-differential equations in the COMSOL Multiphysics® software.

Read More##### Temesgen Kindo October 5, 2016

Cylindrical coordinates are useful for efficiently solving and postprocessing rotationally symmetric problems. The COMSOL Multiphysics® software has built-in support for cylindrical coordinates in the axisymmetry physics interfaces. When defining custom partial differential equations (PDEs) using the mathematical interfaces, paying close attention to their meaning is important. The PDE interfaces assume partial differentiation in a Cartesian system, requiring manual coordinate transformations to change to a cylindrical system. See how to account for such coordinate transformations when using your own PDEs.

Read More##### Bridget Cunningham October 4, 2016

Environmental demands for greater fuel efficiency and lower emissions have sparked an interest in finding an alternative to traditional spark- and compression-ignition engines. While homogeneous charge compression ignition (HCCI) engines offer a viable solution, significant challenges like maintaining control of ignition timing still remain. With simulation tools like the COMSOL Multiphysics® software, you can analyze the combustion process of an HCCI engine and gain deeper insight into ways to advance ignition control.

Read More##### Linus Andersson September 29, 2016

Perforations, in mufflers for example, enable partial sound transmission between chambers as well as in and out of pipes. When simulating perforates, it’s possible to draw and mesh each hole, but this increases the time it takes to solve the model. For a more efficient approach, we can apply a semitransparent boundary. Here, we’ll discuss several techniques for doing so as well as describe a method for computing the transfer impedance of the perforate.

Read More##### Caty Fairclough September 28, 2016

Imagine going to a hospital and having your vitals checked by a machine with a silicon skin, or exploring hard-to-reach areas on the ocean floor with a robot that moves like an octopus. Thanks to soft robotics, a field that involves the design of soft and nonrigid robots, these scenarios may become a possibility in the future. Find out more about this growing technology and the range of opportunities it provides.

Read More##### Björn Bretz September 27, 2016

To help optimize your modeling processes, we are continuously striving to enhance the quality of our meshing capabilities. The recent improvements to the algorithm for generating tetrahedral meshes in the COMSOL Multiphysics® software are one such example. Follow along as we guide you through the process of generating a tetrahedral mesh to highlight this improved functionality and its correlating features, while discussing its role in helping you obtain better simulation results.

Read More##### Caty Fairclough September 26, 2016

The fans in an airplane’s turbofan engine are one of its main sources of noise. In excess, this can cause a range of health problems, including hearing impairment, sleep disturbance, and stress-related illnesses. To optimize the design of turbofan engines to reduce noise pollution and its correlating side effects, you can turn to acoustic modeling. Our jet pipe tutorial model speaks to the benefits of using such an approach.

Read More##### Christopher Boucher September 22, 2016

In the previous installment of this series, we explained two concepts needed to model the release and propagation of real-world charged particle beams. We first introduced probability distribution functions in a purely mathematical sense and then discussed a specific type of distribution — the transverse phase space distribution of a charged particle beam in 2D. Now, let’s combine what we’ve learned and find out how to sample the initial positions and velocities of 3D beam particles from this distribution.

Read More##### Yosuke Mizuyama September 21, 2016

The Gaussian beam is recognized as one of the most useful light sources. To describe the Gaussian beam, there is a mathematical formula called the paraxial Gaussian beam formula. Today, we’ll learn about this formula, including its limitations, by using the Electromagnetic Waves, Frequency Domain interface in the COMSOL Multiphysics® software. We’ll also provide further detail into a potential cause of error when utilizing this formula. In a later blog post, we’ll provide solutions to the limitations discussed here.

Read More##### Amelia Halliday September 20, 2016

To optimize your modeling processes, there are a number of built-in materials available for you to use in the COMSOL Multiphysics® software. Along with these materials are features and functionality that allow you to efficiently assign materials to geometric entities in your model. These tools help expedite the process of assigning materials, specifying material properties, and even comparing the impact of different materials on your simulation results. Here, we’ll highlight three tutorial videos that showcase how to use such tools.

Read More##### Christopher Boucher September 19, 2016

Previously in our Phase Space Distributions in Beam Physics series, we introduced probability distribution functions (PDFs) and various ways to sample from them in the COMSOL Multiphysics® software. Such knowledge of PDFs is necessary to understand how ion and electron beams propagate within real-world systems. In this installment, we’ll discuss the concepts of phase space and emittance as they apply to the release of ions or electrons in beams.

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