LaplacianPDETerm
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LaplacianPDETerm
Details
- The Laplacian term is a differential operator term used to describe many physical phenomena, such as potentials, heat transfer, acoustics, structural mechanics and fluid dynamics.
- LaplacianPDETerm returns a differential operators term to be used as a part of partial differential equations:
- LaplacianPDETerm can be used to model Laplacian equations with dependent variable , independent variables and time variable .
- Stationary model variables vars are vars={u[x1,…,xn],{x1,…,xn}}.
- Time-dependent model variables vars are vars={u[t,x1,…,xn],{x1,…,xn}} or vars={u[t,x1,…,xn],t,{x1,…,xn}}.
- The Laplacian is realized as a DiffusionPDETerm with as a diffusion coefficient resulting in .
- The following parameters pars can be given:
-
parameter default symbol "RegionSymmetry" None - A possible choice for the parameter "RegionSymmetry" is "Axisymmetric".
- "Axisymmetric" region symmetry represents a truncated cylindrical coordinate system where the cylindrical coordinates are reduced by removing the angle variable as follows:
-
dimension reduction equation 1D 2D - The diffusion coefficient affects the meaning of NeumannValue.
Examples
open allclose allBasic Examples (2)Summary of the most common use cases
Scope (3)Survey of the scope of standard use cases
Set up a time-dependent Laplacian:
https://wolfram.com/xid/0do8lvokwhaw6tqxm-v48g2c
Define a 2D axisymmetric stationary Laplacian term:
https://wolfram.com/xid/0do8lvokwhaw6tqxm-jn5edm
Activate the term:
https://wolfram.com/xid/0do8lvokwhaw6tqxm-daelxt
Verify that the axisymmetric case is a consequence of using a truncated cylindrical coordinate system using the operators that compose the Laplacian term:
https://wolfram.com/xid/0do8lvokwhaw6tqxm-b7aov6
Find eigenvalues and vectors of a Laplacian term in a 3D shell:
https://wolfram.com/xid/0do8lvokwhaw6tqxm-m1i2hq
https://wolfram.com/xid/0do8lvokwhaw6tqxm-ypshu3
Applications (1)Sample problems that can be solved with this function
Solve the Laplace equation for inside a finite cylinder of radius 5. The region of analysis is a 3D solid cylinder, but since the region is rotationally symmetric around the axis, an axisymmetric stationary LaplacianPDETerm can be used, and so the region can be defined with truncated cylindrical coordinates in 2D .
https://wolfram.com/xid/0do8lvokwhaw6tqxm-b60lub
https://wolfram.com/xid/0do8lvokwhaw6tqxm-brlabt
Solve the equation and measure time and memory needed to do so:
https://wolfram.com/xid/0do8lvokwhaw6tqxm-97o8d
Visualize the axisymmetric result:
https://wolfram.com/xid/0do8lvokwhaw6tqxm-u6kvl
Print the total time to do the computation and number of megabytes used during the evaluation:
https://wolfram.com/xid/0do8lvokwhaw6tqxm-k7k1yy
The computational cost in time and memory to solve the 2D axisymmetric PDE model is much less than the full 3D model. Create the full 3D region:
https://wolfram.com/xid/0do8lvokwhaw6tqxm-ju8gnc
https://wolfram.com/xid/0do8lvokwhaw6tqxm-i5qrgt
Solve the full 3D model while at the same time improving the boundary position for optimal approximation of the symbolic region and a refinement in the critical area:
https://wolfram.com/xid/0do8lvokwhaw6tqxm-cckrvx
Print the total time to do the computation and number of megabytes used during the evaluation:
https://wolfram.com/xid/0do8lvokwhaw6tqxm-eh2x04
Visualize the result from the 3D solution:
https://wolfram.com/xid/0do8lvokwhaw6tqxm-f7dg22
Compare the timings and memory measurements for the two cases:
https://wolfram.com/xid/0do8lvokwhaw6tqxm-u4w9vh
Visualize the difference between the two solutions:
https://wolfram.com/xid/0do8lvokwhaw6tqxm-ffvjr1
Possible Issues (1)Common pitfalls and unexpected behavior
Neat Examples (1)Surprising or curious use cases
Find your way through a maze by solving a Laplace equation. Set up a maze:
https://wolfram.com/xid/0do8lvokwhaw6tqxm-31w5vk
Convert the graph of the maze to a mesh:
https://wolfram.com/xid/0do8lvokwhaw6tqxm-bb42hk
Solve a Laplace equation with the entrance as a DirichletCondition of value 1 and the exit as a DirichletCondition with value 0. Find the region bounds to set the position of the DirichletCondition:
https://wolfram.com/xid/0do8lvokwhaw6tqxm-3iyocr
https://wolfram.com/xid/0do8lvokwhaw6tqxm-6iqvjp
Visualize the path through the maze:
https://wolfram.com/xid/0do8lvokwhaw6tqxm-54ry6l
Wolfram Research (2020), LaplacianPDETerm, Wolfram Language function, https://reference.wolfram.com/language/ref/LaplacianPDETerm.html (updated 2022).
Text
Wolfram Research (2020), LaplacianPDETerm, Wolfram Language function, https://reference.wolfram.com/language/ref/LaplacianPDETerm.html (updated 2022).
Wolfram Research (2020), LaplacianPDETerm, Wolfram Language function, https://reference.wolfram.com/language/ref/LaplacianPDETerm.html (updated 2022).
CMS
Wolfram Language. 2020. "LaplacianPDETerm." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/LaplacianPDETerm.html.
Wolfram Language. 2020. "LaplacianPDETerm." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/LaplacianPDETerm.html.
APA
Wolfram Language. (2020). LaplacianPDETerm. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LaplacianPDETerm.html
Wolfram Language. (2020). LaplacianPDETerm. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LaplacianPDETerm.html
BibTeX
@misc{reference.wolfram_2024_laplacianpdeterm, author="Wolfram Research", title="{LaplacianPDETerm}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/LaplacianPDETerm.html}", note=[Accessed: 07-January-2025
]}
BibLaTeX
@online{reference.wolfram_2024_laplacianpdeterm, organization={Wolfram Research}, title={LaplacianPDETerm}, year={2022}, url={https://reference.wolfram.com/language/ref/LaplacianPDETerm.html}, note=[Accessed: 07-January-2025
]}