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Related Concept Videos

Implicit Differentiation01:25

Implicit Differentiation

In classical mechanics, motion is often described through relationships between spatial coordinates and time. A car moving along a straight highway with constant acceleration serves as a simple case where velocity is an explicit function of time. This scenario results in a linear equation, enabling straightforward analysis using basic differentiation techniques.In contrast, a satellite in circular orbit follows a path defined by an implicit function. The position of the satellite is constrained...
Improper Integrals: Discontinuous Integrands01:28

Improper Integrals: Discontinuous Integrands

Evaluating Areas Under Curves with DiscontinuitiesA definite integral is considered improper when the integrand is discontinuous at one of the limits of integration. This occurs when the function is undefined or becomes infinite at an endpoint, making the corresponding region under the curve unbounded. Such behavior is commonly associated with vertical asymptotes at the boundary of the interval. To properly define and evaluate these integrals, a limiting process is used to determine whether a...
Implicit Differentiation: Problem Solving01:29

Implicit Differentiation: Problem Solving

Curves defined implicitly, where variables cannot be separated algebraically, require specialized techniques for analysis. The conchoid of Nicomedes exemplifies such a case. Its equation links x and y in a way that prevents isolation of one variable, making implicit differentiation essential to determine the slope and behavior at any point on the curve.The implicit form of the conchoid can be expressed as:To differentiate this equation, y is treated as a function of x, and the chain rule is...
Velocity and Position by Integral Method01:13

Velocity and Position by Integral Method

If acceleration as a function of time is known, then velocity and position functions can be derived using integral calculus. For constant acceleration, the integral equations refer to the first and second kinematic equations for velocity and position functions, respectively.
Consider an example to calculate the velocity and position from the acceleration function. A motorboat is traveling at a constant velocity of 5.0 m/s when it starts to decelerate to arrive at the dock. Its acceleration is...
Continuous Charge Distributions01:17

Continuous Charge Distributions

Imagine a bucket of water. It contains many molecules, of the order of 1026 molecules. Thus, although it contains discrete elements (molecules) at the microscopic level, macroscopically, it can be considered continuous. Small volume elements of water, infinitesimal compared to the bulk of the bucket's volume, still contain many molecules. Under this framework, quantized matter is approximated as continuous for practical purposes.
The electric charge can also be subjected to an analogical...
Approximate Integration01:24

Approximate Integration

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Related Experiment Videos

Asynchronous variational integration using continuous assumed gradient elements.

Sebastian Wolff1, Christian Bucher

  • 1Dynardo Austria GmbH, Wagenseilgasse 14, 1120 Wien, Austria.

Computer Methods in Applied Mechanics and Engineering
|April 2, 2013
PubMed
Summary
This summary is machine-generated.

Asynchronous variational integration (AVI) enhances numerical efficiency for explicit time stepping in finite element analysis by using domain-specific time steps. This method ensures long-term stability and provides a recipe for critical time step estimation.

Keywords:
Asynchronous variational integratorsContinuous assumed gradientCritical time stepSmoothed finite element method

Related Experiment Videos

Area of Science:

  • Computational mechanics
  • Numerical analysis
  • Finite element methods

Background:

  • Explicit time stepping schemes in finite element analysis can be computationally expensive, especially with locally refined meshes.
  • Achieving long-term stability in numerical simulations is crucial for accurate results.
  • Existing methods may lack efficiency when dealing with complex mesh geometries.

Purpose of the Study:

  • To present a modified asynchronous variational integration (AVI) method for finite element analysis.
  • To improve the numerical efficiency of explicit time stepping schemes.
  • To ensure long-term stability in simulations with local spatial refinement.

Main Methods:

  • Implementation of AVI within the finite element framework using a weakened weak form (W2).
  • Association of individual time step lengths to each spatial domain for improved efficiency.
  • Utilizing continuous assumed gradient elements for the numerical examples.

Main Results:

  • Demonstration of enhanced numerical efficiency compared to traditional explicit schemes.
  • Validation of long-term stability through the variational structure of AVI.
  • Provision of practical implementation notes and a method for critical time step estimation.

Conclusions:

  • Modified AVI offers a significant improvement in computational efficiency for explicit finite element methods.
  • The variational structure inherently provides long-term stability.
  • The presented approach is suitable for finite element meshes with local spatial refinement.