Implementing the Weak Form with a COMSOL® App

Chien Liu April 16, 2015

Previously in our weak form series, we discretized the weak form equation to obtain a matrix equation to solve for the unknown coefficients in our simple example problem. Following the same procedure as in this previous blog post, we will implement the equation in the COMSOL Multiphysics® software with additional steps included to examine the matrices. We will find it more convenient to use a COMSOL® software application to display all relevant matrices at once, arranged logically on one screen.

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Chien Liu April 1, 2015

Over half a century ago, Mark Kac gave an interesting lecture on a question that he had heard from Professor Bochner ten years earlier: “Can one hear the shape of a drum?” He focused on the (then undetermined) uniqueness of the set of eigenvalues given the shape of a vibrating membrane. The eigenvalue problem has since been solved and here we explore the “hearing” part of the question by considering some interesting physical effects.

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Chien Liu February 9, 2015

This post continues our blog series on the weak formulation. In the previous post, we implemented and solved an exemplary weak form equation in the COMSOL Multiphysics software. The result was validated with simple physical arguments. Today, we will start to take a behind-the-scenes look at how the equations are discretized and solved numerically.

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Chien Liu January 6, 2015

This blog post is part of a series aimed at introducing the weak form with minimal prerequisites. In the first blog post, we learned about the basic concepts of the weak formulation. All equations were left in the analytical form. Today, we will implement and solve the equations numerically using the COMSOL Multiphysics simulation software. You are encouraged to follow the steps with a working copy of the COMSOL software.

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Chien Liu November 19, 2014

This is an introduction to the weak form for those of us who didn’t grow up using finite element analysis and vector calculus in our daily lives, but are nevertheless interested in learning about the weak form, with the help of some physical intuition and basic calculus.

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Wei Guo July 30, 2014

We have all experienced the boredom and frustration of being stuck in a traffic jam. Very often, traffic congestion comes and goes for no obvious reason. Employing the analogy to gas dynamics, we can now simulate traffic flow using the equation-based modeling capabilities of COMSOL Multiphysics and gain a better understanding of why congestion happens.

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Fabrice Schlegel May 30, 2014

Most numerical simulation methods (finite elements, finite volumes, and finite differences) require stabilization methods when modeling transport applications driven mainly by convection rather than diffusion. With the finite element method (FEM), stabilization means adding a small amount of artificial diffusion. This leads to more robust and faster computational performance. Here, we provide insight on the impact of stabilization on your numerical model. We also look at an alternative numerical method that is very efficient and does not require any stabilization.

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Walter Frei April 30, 2014

We all know that COMSOL Multiphysics can take partial derivatives. After all, it solves partial differential equations via the finite element method. Did you know that you can also solve integrals? That alone shouldn’t be very surprising, since solving finite element problems requires that you integrate functions. The COMSOL software architecture allows you to do a bit more than just evaluate an integral; you can also solve problems where you don’t know the limits of the integral! Here’s how.

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Bettina Schieche April 29, 2014

If you use finite element simulation software, such as COMSOL Multiphysics, you will come across the expression “weak form” at some point. When you do, you may wonder what this expression means. Weak form is actually a very powerful concept. Here, you will learn about its basic ideas and corresponding benefits.

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Bjorn Sjodin January 14, 2014

Many of our users are well aware of the fact that COMSOL Multiphysics can be used to solve partial differential equations (PDEs) as well as ordinary differential equations (ODEs) and initial value problems. It may be less obvious that you can also solve algebraic and even transcendental equations, or in other words, find roots of nonlinear equations in one or more variables with no derivatives in them. Are there real applications for this? Absolutely!

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David Kan December 18, 2013

A prospective user of COMSOL approached me about modeling viscous fingering, which is an effect seen in porous media flow. He hadn’t found a satisfying solution elsewhere, so he turned to COMSOL. I’d like to share with you some of my insight on how to go from idea to model to simulation by taking a “do-it-yourself approach” and utilizing the equation-based modeling capabilities of COMSOL Multiphysics.

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