## Image Denoising and Other Multidimensional Variational Problems

##### 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.

Lire la Suite##### 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®.

Lire la Suite##### 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.

Lire la Suite##### 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.

Lire la Suite##### 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.

Lire la Suite##### 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.

Lire la Suite##### 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.

Lire la Suite##### Siva Sashank Tholeti August 22, 2018

Sparging, a mass transfer process between a gas and a liquid, is commonly encountered in industrial applications such as beverage carbonation and photobioreactors, or even at home for aeration in aquariums. In this blog post, we detail how to model a type of sparging, carbonation, using the COMSOL Multiphysics® software.

Lire la Suite##### Mats Nigam August 20, 2018

When setting up fluid flow simulations, we typically focus on a single (possibly a few) components in a larger system, such as a pump or sedimentation tank in a water treatment plant. Naturally, this raises a question: At what distance can we apply boundary conditions without interfering with the process? In this blog post, we look at the effects of the proximity of inlet and outlet boundaries for interior and exterior flows of a homogeneous fluid with negligible compressibility.

Lire la Suite##### Ravi Ranjan July 20, 2018

While working with rotating components, stability analysis is critical, as instability can lead to catastrophic failure. Rotating systems can lead to unstable responses due to asymmetrical inertia of the disk, asymmetrical stiffness of the shaft, or cross-coupling effects due to bearings. From the designer’s point of view, it’s important to ensure that the potentially unstable modes lie outside the operating range of the machine. Let’s explore how to predict the instability in rotor systems using the COMSOL Multiphysics® software.

Lire la Suite##### Nathan Martin July 17, 2018

Following up on a previous blog post about glacier flow modeling, we are going to delve a bit further into a crucial component of geophysics modeling in general: parameterizing numerical models using observations. Let’s see how we can quantify sensitivity and infer unknown parameters through indirect observations using the COMSOL Multiphysics® software and add-on Optimization Module.

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