Accurate numerical solutions of 2-D elastodynamics problems using compact high-order stencils

2020

Arch Appl Mech

Self Cite

A new numerical approach for the time-dependent wave and heat equations as well as for the timeindependent Poisson equation on irregular domains has been developed. Trivial Cartesian meshes and simple 9-point stencil equations with unknown coefficients are used for 2-D irregular domains. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy. The treatment of the Dirichlet and Neumann boundary conditions in the new approach is related to the development of high-order boundary conditions with the stencils that include the same or a smaller number of grid points compared to that for the regular 9-point internal stencils. At similar 9-point stencils, the accuracy of the new approach is two orders higher than that for the linear finite elements. The numerical results for irregular domains in Part 2 of the paper also show that at the same number of degrees of freedom, the new approach is even much more accurate than the quadratic and cubic finite elements with much wider stencils. Similar to our recent results on regular domains, the order of the accuracy of the new approach for the Poisson equation on irregular domains with square Cartesian meshes is higher than that with rectangular Cartesian meshes. The new approach can be directly applied to other partial differential equations.

Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

2020

Arch Appl Mech

Self Cite

“…Next, the test problem for the trapezoidal plate (see Figure 5A) with the Dirichlet boundary conditions along the entire boundary is solved by OLTEM and by conventional finite elements. The exact solution is given by Equation (36). Figure 6 shows the distribution of the exact solution for the displacements u and v as well as the distribution of the relative errors in displacements e u and e v of the numerical solution obtained by OLTEM on the square (b y = 1) Cartesian mesh of size h = 1∕20.…”

Section: The Dirichlet Boundary Conditionsmentioning

confidence: 99%

“…Recently, in our papers we have developed a new numerical approach called the optimal local truncation error method. It has been developed for the scalar PDEs on regular and irregular domains (see References 32‐34) as well as for a system of PDEs on regular domains; see References 35,36 for the 2‐D time‐dependent and time‐independent elasticity on regular domains. At the same structure of the semidiscrete or discrete equations, OLTEM provides the optimal order of accuracy that exceeds the order of accuracy of many known numerical approaches on regular and irregular domains.…”

Section: Introductionmentioning

confidence: 99%

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

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.

“…Optimal local truncation error method (OLTEM) for the solution of PDEs with constant coefficients on regular and irregular domains with Cartesian meshes has been recently developed in our papers (Idesman, 2020;Dey and Idesman, 2020;Idesman and Dey, 2019;Idesman and Dey, 2020c;Idesman and Dey, 2020b;Idesman and Dey, 2020a;Idesman and Dey, 2020d). At the same structure of the semidiscrete or discrete equations, OLTEM provides the optimal order of accuracy that exceeds the order of accuracy of many known HFF 32,8 numerical approaches on regular and irregular domains.…”

Section: Introductionmentioning

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.