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

Numerical Meth Engineering

Dey

2022

Recently we have developed the optimal local truncation error method (OLTEM) for scalar PDEs on irregular domains and unfitted Cartesian meshes. Here, OLTEM is extended to a much more general case of a system of PDEs for the 2-D time-independent elasticity equations on irregular domains. Compact 9-point uniform and nonuniform stencils (with the computational costs of linear finite elements) are used with OLTEM. The stencil coefficients are assumed to be unknown and are calculated by the minimization of the local truncation error.It is shown that the second order of accuracy is the maximum possible accuracy for 9-point stencils independent of the numerical technique used for their derivations. The special treatment of the Neumann boundary conditions has been developed that does not increase the size of the stencils. The numerical examples are in agreement with the theoretical findings. They also show that due to the minimization of the local truncation error, OLTEM is much more accurate than linear finite elements and than quadratic finite elements (up to engineering accuracy of 0.1%-1%) at the same numbers of degrees of freedom. Due to the computational efficiency and trivial unfitted Cartesian meshes that are independent of irregular domains, the proposed technique with no remeshing for the shape change of irregular domains will be effective for many engineering applications.

Section: Introductionmentioning

confidence: 82%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Optimal local truncation error method to solution of 2‐D time‐independent elasticity problems with optimal accuracy on irregular domains and unfitted Cartesian meshes

Numerical Meth Engineering

Dey

2022

“…The stencils coefficients of the cut stencils can be derived by zeroing k i = 0 ( i is the number of the grid point cut by the interface). Then, the stencil coefficients and the corresponding local truncation errors of the cut stencils can be analytically found similar to the procedure for the non-cut stencils (Idesman, 2020; Idesman and Dey, 2020). The orders of the local truncation error of the cut stencils are 4 for the 8-point cut stencils and 14 for the 24-point cut stencils (these orders are the same or higher compared to those considered below for the stencils with interfaces).…”

Section: Optimal Local Truncation Error Methods For the Two-dimension...mentioning

confidence: 99%

3rd and 11th orders of accuracy of ‘linear’ and ‘quadratic’ elements for Poisson equation with irregular interfaces on Cartesian meshes

Dey

2021

HFF

Purpose The purpose of this paper is as follows: to significantly reduce the computation time (by a factor of 1,000 and more) compared to known numerical techniques for real-world problems with complex interfaces; and to simplify the solution by using trivial unfitted Cartesian meshes (no need in complicated mesh generators for complex geometry). Design/methodology/approach This study extends the recently developed optimal local truncation error method (OLTEM) for the Poisson equation with constant coefficients to a much more general case of discontinuous coefficients that can be applied to domains with different material properties (e.g. different inclusions, multi-material structural components, etc.). This study develops OLTEM using compact 9-point and 25-point stencils that are similar to those for linear and quadratic finite elements. In contrast to finite elements and other known numerical techniques for interface problems with conformed and unfitted meshes, OLTEM with 9-point and 25-point stencils and unfitted Cartesian meshes provides the 3-rd and 11-th order of accuracy for irregular interfaces, respectively; i.e. a huge increase in accuracy by eight orders for the new 'quadratic' elements compared to known techniques at similar computational costs. There are no unknowns on interfaces between different materials; the structure of the global discrete system is the same for homogeneous and heterogeneous materials (the difference in the values of the stencil coefficients). The calculation of the unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of OLTEM at a given stencil width. The numerical results with irregular interfaces show that at the same number of degrees of freedom, OLTEM with the 9-points stencils is even more accurate than the 4-th order finite elements; OLTEM with the 25-points stencils is much more accurate than the 7-th order finite elements with much wider stencils and conformed meshes. Findings The significant increase in accuracy for OLTEM by one order for 'linear' elements and by 8 orders for 'quadratic' elements compared to that for known techniques. This will lead to a huge reduction in the computation time for the problems with complex irregular interfaces. The use of trivial unfitted Cartesian meshes significantly simplifies the solution and reduces the time for the data preparation (no need in complicated mesh generators for complex geometry). Originality/value It has been never seen in the literature such a huge increase in accuracy for the proposed technique compared to existing methods. Due to a high accuracy, the proposed technique will allow the direct solution of multiscale problems without the scale separation.

Optimal local truncation error method for 2‐D elastodynamics problems on irregular domains and unfitted Cartesian meshes

Num Anal Meth Geomechanics

Dey²

2022

The optimal local truncation error method (OLTEM) has been developed for 2‐D time‐dependent elasticity problems on irregular domains and trivial unfitted Cartesian meshes. Compact nine‐point uniform and nonuniform stencils (similar to those for linear finite elements on uniform meshes) are used with OLTEM. The stencil coefficients are assumed to be unknown and are calculated by the minimization of the local truncation error. It is shown that the second order of accuracy is the maximum possible accuracy for nine‐point stencils independent of the numerical technique used for their derivations. The special treatment of the Neumann boundary conditions has been developed that does not increase the size of the stencils. The cases of the nondiagonal and diagonal mass matrices are considered for OLTEM. The results of numerical examples are in agreement with the theoretical findings. They also show that due to the minimization of the local truncation error, OLTEM with the nondiagonal mass matrix is much more accurate than linear finite elements and than quadratic and cubic finite elements (up to the engineering accuracy of 0.1%–1%) at the same numbers of degrees of freedom. The proposed numerical technique can be efficiently used for many engineering applications including geomechanics.