Journal ArticleDOI
A new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes—part 2: numerical simulations and comparison with FEM
B. Dey,Alexander Idesman +1 more
Reads0
Chats0
TLDR
In this article, a new numerical approach for the time-dependent wave and heat equations as well as for time-independent Poisson equation developed in Part 1 is applied to the simulation of 1-D and 2-D test problems on regular and irregular domains.Abstract:
A new numerical approach for the time-dependent wave and heat equations as well as for the time-independent Poisson equation developed in Part 1 is applied to the simulation of 1-D and 2-D test problems on regular and irregular domains. Trivial conforming and non-conforming Cartesian meshes with 3-point stencils in the 1-D case and 9-point stencils in the 2-D case are used in calculations. The numerical solutions of the 1-D wave equation as well as the 2-D wave and heat equations for a simple rectangular plate show that the accuracy of the new approach on non-conforming meshes is practically the same as that on conforming meshes for both the Dirichlet and Neumann boundary conditions. Moreover, very small distances (
$$0.1 h - 10^{-9}h$$
where h is the grid size) between the grid points of a Cartesian mesh and the boundary do not decrease the accuracy of the new technique. The application of the new approach to the 2-D problems on an irregular domain shows that the order of accuracy is close to four for the wave and heat equations and is close to five for the Poisson equation. This is in agreement with the theoretical results of Part 1 of the paper. The comparison of the numerical results obtained by the new approach and by FEM shows that at similar 9-point stencils, the accuracy of the new approach on irregular domains is two orders higher for the wave and heat equations and three orders higher for the Poisson equation than that for the linear finite elements. Moreover, the new approach yields even much more accurate results than the quadratic and cubic finite elements with much wider stencils. An example of a problem with a complex irregular domain that requires a prohibitively large computation time with the finite elements but can be easily solved with the new approach is presented.read more
Citations
More filters
Journal Article
A Cartesian grid embedded boundary method for solving the Poisson and heat equations with discontinuous coefficients in three dimensions
TL;DR: In this paper, a Cartesian cut-cell/embedded boundary method is used to represent the interface between materi- als, as described in Johansen & Colella (1998).
Journal ArticleDOI
A new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes—Part 1: the derivations for the wave, heat and Poisson equations in the 1-D and 2-D cases
TL;DR: In this paper, a new numerical approach for the time dependent wave and heat equations as well as for time independent Poisson equation on irregular domains has been developed for 2-D irregular domains.
Journal ArticleDOI
Point cloud-based elastic reverse time migration for ultrasonic imaging of components with vertical surfaces
TL;DR: In this paper, a point cloud-based elastic reverse time migration (PC-based ERTM) method is proposed for non-destructive evaluation of components with vertical or steeply dipping surfaces and demonstrates its ability of accurately characterizing multiple defects hidden in the interior of the component based on a limited coverage of ultrasonic linear phased array.
Journal ArticleDOI
Point cloud-based elastic reverse time migration for ultrasonic imaging of components with vertical surfaces
TL;DR: In this article , a point cloud-based elastic reverse time migration (PC-based ERTM) method is proposed for non-destructive evaluation of components with vertical or steeply dipping surfaces and demonstrates its ability of accurately characterizing multiple defects hidden in the interior of the component based on a limited coverage of ultrasonic linear phased array.
Journal ArticleDOI
A high-order numerical approach with Cartesian meshes for modeling of wave propagation and heat transfer on irregular domains with inhomogeneous materials
Alexander Idesman,Bikash Dey +1 more
TL;DR: In this article, the authors proposed a new numerical approach for PDEs with constant coefficients on irregular domains and Cartesian meshes, based on the representation of the stencil coefficients as functions of the mesh size.
References
More filters
ReportDOI
Code Verification by the Method of Manufactured Solutions
Kambiz Salari,Patrick Knupp +1 more
TL;DR: It is shown that MMS can be applied to a variety of engineering codes which numerically solve partial differential equations, and its principle advantage is that code capabilities are tested in full generality.
Journal ArticleDOI
A Cartesian Grid Embedded Boundary Method for Poisson's Equation on Irregular Domains
Hans Johansen,Phillip Colella +1 more
TL;DR: A numerical method for solving Poisson's equation, with variable coefficients and Dirichlet boundary conditions, on two-dimensional regions using a finite-volume discretization, which embeds the domain in a regular Cartesian grid.
Journal ArticleDOI
Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method
Erik Burman,Peter Hansbo +1 more
TL;DR: A fictitious domain method where the mesh is cut by the boundary where the primal solution is computed only up to the boundary and the solution itself is defined also by nodes outside the domain.
Journal ArticleDOI
A Cartesian grid embedded boundary method for hyperbolic conservation laws
TL;DR: A second-order Godunov algorithm to solve time-dependent hyperbolic systems of conservation laws on irregular domains by hybridizing the authors' conservative discretization with a stable, nonconservative discretized at irregular control volumes, and redistributing the difference in the mass increments to nearby cells in a way that preserves stability and local conservation.
Journal ArticleDOI
Geometric modeling, isogeometric analysis and the finite cell method
TL;DR: The possibility to directly couple the finite cell method to CSG, without any necessity for meshing the three-dimensional domain, is discussed, and a combination of the best of the two approaches IGA and FCM is explored, closely following ideas of the recently introduced shell FCM.
Related Papers (5)
A Cartesian grid embedded boundary method for Poisson`s equation on irregular domains
Hans Johansen,Phillip Colella +1 more