Related Experiment Video
Updated: May 28, 2026
11:28
A Coupled Experiment-finite Element Modeling Methodology for Assessing High Strain Rate Mechanical Response of Soft Biomaterials
Published on: May 18, 2015
Dual Formulations of Mixed Finite Element Methods with Applications
Andrew Gillette1, Chandrajit Bajaj
1Department of Mathematics, University of Texas at Austin.
Computer Aided Design
|October 11, 2011
Summary
The choice of discrete Hodge star is crucial for the numerical stability of mixed finite element methods. New methods can be developed using interpolation functions and Hodge stars on dual meshes.
Area of Science:
- Numerical analysis
- Computational mathematics
- Scientific computing
Background:
- Mixed finite element methods solve partial differential equations (PDEs) using multiple variables.
- Discrete Exterior Calculus provides a theoretical framework for these methods.
- Degrees of freedom are typically stored on primal and dual meshes with a discrete Hodge star for information transfer.
Purpose of the Study:
- To analyze the impact of the discrete Hodge star on numerical stability in mixed finite element methods.
- To introduce new mixed methods by defining interpolation functions and discrete Hodge stars on dual meshes.
Main Methods:
- Theoretical analysis of mixed finite element methods.
- Development of interpolation functions and discrete Hodge stars for dual meshes.
- Application and examination of methods in magnetostatics and Darcy flow.
Main Results:
- The selection of the discrete Hodge star significantly influences the numerical stability of mixed methods.
- New mixed finite element methods were defined using novel interpolation and Hodge star constructions on dual meshes.
Conclusions:
- The discrete Hodge star is a critical component for ensuring the stability of mixed finite element methods.
- The proposed methods offer new possibilities for solving problems in fields like magnetostatics and fluid dynamics.
Related Concept Videos
01:29Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving
Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
01:22Fundamental Theorem of Calculus I: Problem Solving
In many engineering and environmental applications, accumulated quantities are determined from rates that vary over time. A common example arises in water management, where a supply system pumps water into a storage tank at a rate that changes with time. Accurately determining how much water has entered the tank over a given period is essential for maintaining proper pressure, scheduling operations, and ensuring system safety.The flow rate of water into the tank is described by a time-dependent...
01:23Stresses under Combined Loadings
When analyzing a bent tube with a circular cross-section subjected to multiple forces, it is crucial to determine the stress distribution in order to maintain structural integrity under varied load conditions.
The process begins by slicing the tube at critical points and analyzing the internal forces and stress components at these sections, focusing on the centroid. Normal stresses, generated by axial forces and bending moments, are either compressive or tensile and vary across the section from...
The process begins by slicing the tube at critical points and analyzing the internal forces and stress components at these sections, focusing on the centroid. Normal stresses, generated by axial forces and bending moments, are either compressive or tensile and vary across the section from...
01:11Bending of Members Made of Several Materials
In analyzing a structural member composed of two different materials with identical cross-sectional areas, it is crucial to understand how their distinct elastic properties affect the member's response under load. The analysis involves assessing stress and strain distributions using the transformed section concept, which accounts for variations in material properties.
Hooke's Law determines stress in each material, stating that stress is proportional to strain but varies due to each material's...
Hooke's Law determines stress in each material, stating that stress is proportional to strain but varies due to each material's...
01:29Two-Dimensional Force System: Problem Solving
Solving problems related to two-dimensional force systems is an essential aspect of mechanics and engineering. By applying the principles of vector analysis and force equilibrium, one can determine the effect of multiple forces acting on an object in a two-dimensional space.
The first step to solving a two-dimensional force system problem is to draw a free-body diagram of the object under consideration. This diagram helps identify all the external forces acting on the object, including their...
The first step to solving a two-dimensional force system problem is to draw a free-body diagram of the object under consideration. This diagram helps identify all the external forces acting on the object, including their...
01:21Couples: Scalar and Vector Formulation
One might wonder how the captain of a large ship can navigate through the ocean with just a turn of the steering wheel. The answer lies in the concept of two parallel forces that are equal in magnitude and opposite sense, creating a couple moment.
A couple moment is a rotational force that tends to rotate the steering wheel. The wheel's rotation can either be in a clockwise or anticlockwise direction. The right-hand rule is a helpful method for determining the direction of a couple moment. To...
A couple moment is a rotational force that tends to rotate the steering wheel. The wheel's rotation can either be in a clockwise or anticlockwise direction. The right-hand rule is a helpful method for determining the direction of a couple moment. To...