## 2D Meshing Issues with SVOffice™ 2009

Presented by: Dr. Murray Fredlund, President of SoilVision Systems Ltd.

### Introduction

Hi, my name is Murray Fredlund. Welcome to the session 2D Geometry - Advanced CAD Drawing Tools within the SVOffice 2009 Next Generation Geotechnical software suite.

### Outline

This presentation is entitled "2D Meshing" and is designed to cover the basic meshing concepts associated with the SVOffice geotechnical software suite. Once this presentation is completed the user should have a clear idea of the primary issues involved in creating a finite element mesh from 2-D geometry. The concepts presented in this presentation are applicable to all of our finite element software products.

This presentation will cover the topics of giving the user a basic overview of meshing, controlling the mesh, creating a mesh, as well as automatic mesh refinement and common meshing problems. One of the primary benefits of the SVOffice software is its powerful mesh generation and refinement algorithms. This presentation aims to give an overview of the common utilization of these algorithms.

### Meshing Overview

The meshing process happens after model geometry, boundary conditions, and material properties have been defined for a model but prior to the solution of the numerical model. It is therefore the step between model design and solution of the finite element equations. The difficulty is that the quality and density of the mesh potentially have a direct bearing on the quality of the solution. Therefore the meshing process is important in realizing a reasonable solution for any finite element model.

The goal of the meshing process is to discretize the problem into small elements for which the governing partial differential equations can be solved. Solving a finite element problem involves first defining the equations which govern the behavior of each element and then solving them simultaneously for the entire modeling domain. It is mathematically assumed that an interpolation function can adequately describe the behavior of the main variable such as head or temperature across a particular element. It is important that the mesh resolution be sufficient to adequately solve the problem at hand.

Traditionally mesh generation has been problematic and difficult. Early solvers required that the user define their own mesh and enter node points manually. This process was ultimately extremely time-consuming and problematic to reasonably solve problems in a consulting timeframe.

One of the key benefits of the SVOffice software is that significant research has been put into the meshing algorithms present in the software. The goal of the SVOffice software package is therefore to allow the user to focus on the concept of the numerical model at hand rather than the meshing problem or the solution of finite element equations.

To achieve this SVOffice implements fully automatic mesh generation as well as fully automatic mesh refinement to make the process of finding a reasonable solution as easy as possible. SVOffice continues to be a market leader in this area as automatic mesh refinement has been offered in the software since 2001. Automatic mesh refinement means that the mesh is automatically refined in crucial areas such that the solution accuracy is increased and convergence problems are alleviated.

The meshing algorithms in SVOffice software are contained in the solver module through a proprietary algorithm. The user is afforded a certain amount of control of the mesh in the front-end but the actual meshing happens during the solution process. The reason that the mesh is linked to the solution process is that the mesh refinement is triggered by areas of the solution where increased accuracy is needed. Therefore the mesh refinement algorithm is coupled together with the solution algorithm.

As part of the basics of the meshing process, the meshing occurs with triangular elements. However these elements may be linear, quadratic, or cubic in nature. Different types of elements are reasonable in different application scenarios.

Part of the meshing process means that the node points are added at all geometry node points such that region boundaries are accurately represented.

It is also worth noting that node points are placed on all plot and flux section locations such that the results presented in the plots are exact reporting of nodal values and not interpolated values.

The meshing algorithm has been proven in hundreds of models over the past number of years and is fast and generally robust. Especially in 2-D models the mesh generation time is generally insignificant.

### Controlling the Mesh

It is important in certain situations that the user be able to change certain options related to the mesh. All of our finite element software packages allow intricate control of the mesh in the Mesh Generation and Refinement dialog. This dialog can be found under the Model > Mesh Generation... menu option.

There are a number of detailed controls for the mesh generation which can be set but the most common questions our company receives are related to the control of the mesh density in regions or along region boundaries. The mesh generation dialog contains detailed controls for setting the mesh spacing in any global manner as well as for individual regions. Mesh resolution along any particular shape boundary line can also be specified.

Furthermore it is important that any mesh generation process not result in elements which are adversely skewed. An aspect ratio is therefore provided in the software with which the user can control the maximum aspect ratio of any particular element. This helps greatly in the generation of a quality finite element mesh.

### Creating a Mesh

As previously mentioned the creation of the finite element mesh happens during the solution process in the SVOffice software. It's worthy to note that the mesh creation process converts to a slice generation process for SVSlope and is therefore not applicable for the SVSlope software with the exception of some models which couple in a stress analysis.

The basis of the software is that the user is encouraged to focus on the purpose of the numerical model rather than the finite element mesh. Although details regarding the mesh generation may be found in the theory manual for each product is not important that the user have an in-depth knowledge of the mesh generation process in order to use the software product.

Generally speaking the SVOffice software modules attempt to guide the user in the process of creating geometry which yields a well-developed finite element mesh. Therefore automated algorithms run in the background when the user is creating geometry in order to ensure that entered geometry is consistent with the principles required by the meshing algorithm.

That being said it is not possible to guard against every possible scenario where the user may enter a problematic conceptual model which yields problems in the meshing process.

### Mesh Refinement

It is important to realize that the meshing process is made up of two primary steps, namely i) the initial mesh generation step, and ii) the mesh refinement step. Both steps are important for ensuring a quality finite element mesh which yields a reasonable solution.

SVOffice has implemented automatic mesh refinement since 2001 in its first package and it remains the only geotechnical software suite to implement such a feature. The mesh is refined based on a subdivision process such that the mesh node points never actually move.

The mesh refinement is implemented in all of our 1-D, 2-D, and 3-D finite element packages. Why is this? It's because that "insufficient spatial resolution is commonly recognized to be the primary source of errors in numerical solutions of partial differential equations for water flow and solute transport in the vadose zone" as commented by Yea in 2000. This is particularly important for infiltration of water into a dry soil and the movement of a contaminant front.

### Meshing Difficulties

However even with advanced automatic mesh generation and refinement it is still possible for there to be difficulties in meshing if a model is not described properly. Therefore the following slides illustrate common meshing problems which have been encountered by our customers. It is important to note that it is not possible to guard against every scenario the user could possibly create which might cause meshing problems or violate the principles of the meshing algorithm. The focus of the SVOffice software has been to implement drawing guides such that the possibility of encountering such meshing problems is minimized.

With each meshing difficulty the solution implemented by SVOffice to overcome these problems is also presented.

The first problem which can be encountered is one where regions potentially overlap. This situation might arise if the user inadvertently enters wrong geometry or imports geometry which has been entered poorly in AutoCAD. If the user finds himself with this scenario there are several solutions which can be utilized to fix this scenario.

Firstly there is the use of global snapping where the user can snap all points to a specific grid. If this solution does not work then the user can manually and graphically move a node point to a new location.

If the overlap is intended and the overlapping region is desired to be a new region then the user can use the region intersection tool to calculate intersection points and create a new region.

It should be noted that there is also a news article which has been written on region intersections which is available on the SoilVision systems website at the address shown.

Another potential meshing difficulty is related to too many boundary points on a region. This is a typical scenario with a region that has been generated in AutoCAD from field data and may contain an adversely large number of points.

To handle this scenario there is a specific line simplification algorithm which has been implemented in the software in order to reduce the number of region points to a reasonable number. The user must use their own professional judgment as well to determine what is a reasonable level of complexity of the numerical model which still retains the intended use of the model.

A specific article outlining the line simplification feature of the software may be found at http://www.soilvision.com/features/line-simplification.shtml.

It is also possible that geometry objects have been drawn too close together and often this will not create a specific meshing difficulty but will cause an inadvertently dense mesh. In this scenario the user is encouraged to use local or global snapping to remove small tolerances from the numerical model. This process is done easily under the tools provided on the geometry menu.

It is also possible that the user has entered two adjacent regions with nonmatching adjacent region points. This can create difficulties for the meshing algorithms especially if the regions do not exactly line up. The scenario is typically created when the user has imported geometry from AutoCAD. If geometry has been entered within the SVOffice software there are object snapping algorithms which guide the user such that the scenario does not happen. It is recommended that the user make use of the object snapping tool in order to snap to previously defined node or line segments when drawing new geometry.

### Conclusions

In conclusion this presentation covers mesh creation as well as the primary topics of getting an overview of the meshing algorithms, controlling the mesh, creating a mesh, automatic mesh refinement, and overcoming common meshing problems.