An Application of Isogeometric Analysis and Boundary Integral Element Method for Solving Nonlinear Contact Problems

Round 1

Reviewer 1 Report

Re: A novel application of the boundary integral element method and isogeometric for the solution of nonlinear contact problems

               S.V. Camacho Gutierrez et al

 

The paper presents application of the boundary integral element method combined with the isogeometric boundary description. The contact problem of plane strain compression of two identical cylinders was numerically analysed by the proposed ISO-BEM method and compared with the FEM solution.

1.The paper is written in poor English with numerous misprints and linguistic mistakes requiring correction, including paper title and abstract. The novelty of analysis is claimed in the title, but all elements follow literature presentations with some modifications.

 

2. The unilateral contact conditions are not stated with proper variational background and the description in Sect. 3.2 is not clear. The optimization problem is addressed to determination of the contact zone with application of particle swarm optimization (PSO), but there is no formulation of the objective function nor constraints. This section, presenting some novelty in approach to treatment of contact conditions should be rewritten with more rigour and clarity.

 

4. The results of numerical analysis in Sect.4 are also not clearly presented. The problem treated of two cylinders in contact has analytical solution for linear elastic materials and the numerical solution obtained should refer to this solution, by comparing penetration depth, contact zone size, local stress and displacement fields obtained by the proposed ISO-BEM and FEM methods, referred to exact analysis (cf. K.L. Johnson book).

 

To be publishable , the paper needs essential improvements.

 

 

Author Response

Dear Reviewer:

I appreciate your comments. I've attached my reply below.

 

Regards.

S. V. CAMACHO

Author Response File: Author Response.pdf

Reviewer 2 Report

The manuscript by Camacho Gutiérrez et al. presents a boundary element method that uses isogeometric mapping for improved description of computation geometry applied to contact problems. My main concern is that manuscript is missing some detailed description, particularly in results section. Thus I propose major improvements to the section containing numerical results:

1) How many nodes and boundary elements are used in section 4.3. Authors mention number of control points for description of geometry, but it is not clear how many nodes is used for analysis. Is the number of boundary elements and nodes same as in section 4.2.?

2) The idea of the authors is to use FEM model to obtain a referent solution for comparison with BEM and IGA-BEM. The comparison is only qualitative. I would suggest creating fine-grid FEM solution using even finer grid (and checking solution convergence under mesh refinement). Then by using that referent solution (as “exact” solution) perform quantitative analysis. Statement that IGA-BEM produces “better” results just by comparing figures (especially since both approaches produce different solution then FEM) is not acceptable.

3) Comment order of accuracy (i.e. order of basis functions) used for all methods: FEM, BEM and IGA-BEM.

General comment for comparison between FEM and BEM type of methods: Even that BEM uses generally lower number of degrees of freedom (DOF), it produces dense matrix (considering linear system of discretized equations) as opposite to sparse matrix produced by FEM. Thus computational cost (e.q. CPU time and memory requirements) cannot be directly estimated by comparing number of DOF. The interesting study would be comparing computational cost for reaching same order of accuracy comparing with known referent (analytical of fine-grid) solution. Measuring cpu time would not be appropriate for presented examples because number of DOF is relatively small so even direct solvers (e.g. Gauss elimination) can be pretty efficient. However, it would be interesting to get some insight what can be expected if one tries to solve much bigger problems (e.g. using >10^6 DOF).

Additional comments:

• Title: The word “isogeometric” is used as stand alone which I’m not sure that is appropriate. Suggesting minor reformulation of the title, like using “isogeometric mapping”, “isogeometric technology”, or similar.
• ln 25 – “by applying divergence theorem” is insufficient and inappropriate description for BEM. By just applying divergence theorem would lead to control volume formulation. Mentioning the Green function is more important here, because the BEM cannot be used for problems where Green function is unknown. I recommend adding sentence or two more about general BEM idea and referencing some more classical BEM textbooks.
• ln 40 – correct 2004 to 2005
• ln 40 – I don't think this sentence is appropriate. IGA was not designed to work “with FEM”. IGA in some sense represents new discretization approach, which, when used in conjunction with Galerkin approach can be regarded as NURBS based finite element method, since it uses FEM framework. I recommend slight reformulation.
• ln 42 add word “method” before “(IGA-BEM)”
• ln 115 – capitalize Non-Uniform
• ln 189 - “as suggest [4] from [29].” ?
• Figures 8-9, 11-13 missing color legend
• General comment: comparing IGA vs true IGA-BEM (using NURBS as interpolation functions) in terms of computational efficiency could be very interesting.

 

 

Author Response

Dear Reviewer:

I appreciate your comments. I've attached my reply below.

 

Regards.

S. V. CAMACHO

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

In the revised manuscript authors have included more details and I see improved version of the manuscript. I can recommend manuscript for publication in AS.