## Nonstandard Constraints and the Power of Weak Contributions

##### Temesgen Kindo October 16, 2018

Have you ever wanted to add a certain boundary or domain condition to a physics problem but couldn’t find a built-in feature? Today, we will show you how to implement nonstandard constraints using the so-called weak contributions. Weak contributions are, in fact, what the software internally uses to apply the built-in domain and boundary conditions. They provide a flexible and physics-independent way to extend the applicability of the COMSOL Multiphysics® software.

Read More##### Lipeng Liu October 12, 2018

Using the Numeric Port feature, available in the COMSOL Multiphysics® software with the add-on RF Module, the mode of a port with an arbitrary shape can be computed numerically via a boundary mode analysis. By adding a Frequency Domain or an Adaptive Frequency Sweep study, the S-Parameter and Smith plots can be obtained. The numeric port also enables us to calculate the characteristic impedance of transmission lines operating in the transverse electromagnetic (TEM) mode.

Read More##### Harishanker Gajendran September 27, 2018

In a previous blog series on the weak formulation, my colleague Chien Liu introduced the weak form for stationary problems and the methodology to implement it in the COMSOL Multiphysics® software. This blog post is an extension of the weak form series, showing how to solve a time-dependent problem using the Weak Form PDE interface. As you will see, not much changes in the procedure. Allow us to demonstrate…

Read More##### Jiyoun Munn September 25, 2018

When we analyze high-frequency electromagnetics problems using the finite element method (FEM), we often compute S-parameters in the frequency domain without reviewing the results in the complementary domain; that is, the time domain. The time domain is where we can find other useful information, such as time-domain reflectometry (TDR). In this blog post, we will demonstrate data conversion between two domains in order to efficiently obtain results in the desired computation domain through a fast Fourier transform (FFT) process.

Read More##### Temesgen Kindo September 21, 2018

We previously discussed how to solve 1D variational problems with the COMSOL Multiphysics® software and implement complex domain and boundary conditions using a unified constraint enforcement framework. Here, we extend the discussion to multiple dimensions, higher-order derivatives, and multiple unknowns with what we hope will be an enjoyable example: variational image denoising. We conclude this blog series on variational problems with some recommendations for further study.

Read More##### Temesgen Kindo September 17, 2018

How do you find the shortest overland distance between two points across a lake? Such obstacles and bounds on solutions are often called inequality constraints. Requirements for nonnegativity of gaps between objects in contact mechanics, species concentrations in chemistry, and population in ecology are some examples of inequality constraints. Previously in this series, we discussed equality constraints on variational problems. Today, we will show you how to implement inequality constraints when using equation-based modeling in COMSOL Multiphysics®.

Read More##### Temesgen Kindo September 11, 2018

In the first part of this blog series, we discussed how to solve variational problems with simple boundary conditions. Next, we proceeded to more sophisticated constraints and used Lagrange multipliers to set up equivalent unconstrained problems. Today, we focus on the numerical aspects of constraint enforcement. The method of Lagrange multipliers is theoretically exact, yet its use in numerical solutions poses some challenges. We will go over these challenges and show two mitigation strategies: the penalty and augmented Lagrangian methods.

Read More##### Temesgen Kindo September 7, 2018

In the first part of this blog series, we discussed variational problems and demonstrated how to solve them using the COMSOL Multiphysics® software. In that case, we used simple built-in boundary conditions. Today, we will discuss more general boundary conditions and constraints. We will also show how to implement these boundary conditions and constraints in the COMSOL® software using the same variational problem from Part 1: (the soap film) — and just as much math.

Read More##### Chandan Kumar September 5, 2018

To characterize hyperelastic materials, we need experimental data from a variety of tests, including subjection to uniaxial tension and compression, biaxial tension and compression, and torsion. Here, we show how to model the compression of a sphere made of an elastic foam using tension and compression test data obtained via uniaxial and equibiaxial tests. We demonstrate the use of the compressible Storakers hyperelastic material model for computation as well as how force-versus-stretch relationships are calculated for uniaxial and equibiaxial tests.

Read More##### Temesgen Kindo September 4, 2018

What do soap films, catenary cables, and light beams have in common? They behave in ways that minimize certain quantities. Such problems are prevalent in science and engineering fields such as biology, economics, elasticity theory, material science, and image processing. You can simulate many such problems using the built-in physics interfaces in the COMSOL Multiphysics® software, but in this blog series, we will show you how to solve variational problems using the equation-based modeling features.

Read More##### Henrik Ekström August 29, 2018

Ion-exchange membranes are widely employed within the field of electrochemical engineering. In polymer electrolyte fuel cells and vanadium flow batteries, they are used to conduct ions and at the same time prevent reactants and electrons from crossing between the two flow compartments. The ability to promote the passage of ions of either positive or negative charge is also used in electrodialysis for cleaning water from ions. In this blog post, we will explore the ion-selective capabilities of ion-exchange membranes.

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