ORIGINAL ARTICLE

Volume rendering visualization of 3D spherical mantleconvection with an unstructured mesh

Shi Chen Æ Huai Zhang Æ David A. Yuen ÆShuxia Zhang Æ Jian Zhang Æ Yaolin Shi

Received: 2 March 2008 / Accepted: 30 March 2008 / Published online: 13 June 2008

� Springer-Verlag 2008

Abstract We propose a new approach to utilize the

algorithm of hardware-assisted visibility sorting (HAVS) in

the 3D volume rendering of spherical mantle convection

simulation results over unstructured grid configurations.

We will also share our experience in using three different

spherical convection codes and then taking full advantages

of the enhanced efficiency of visualization techniques,

which are based on the HAVS techniques and related

transfer functions. The transfer function is a powerful tool

designed specifically for editing and exploring large-scale

datasets coming from numerical computation for a given

environmental setting, and generates hierarchical data

structures, which will be used in the future for fast access

of GPU visualization facilities. This method will meet the

coming urgent needs of real-time visualization of 3D

mantle convection, by avoiding the demands of huge

amount of I/O space and intensive network traffic over

distributed parallel terascale or petascale architecture.

Keywords Volume rendering � Visualization �Mantle convection � Unstructured mesh

Introduction

Since the late 1960s (Torrance and Turcotte 1971;

McKenzie et al. 1974) numerical modeling of mantle

convection has developed a rich history. For numerical

simulation of the mantle convective process, the spherical

coordinate system is often employed for global geody-

namical problems, such as, the coupled modeling of global

plate tectonic with mantle convection physics, the multi-

scale investigate of the D00 layer and core–mantle boundary

characteristics, the global 670 km discontinuity boundary

atop the lower mantle, (Gable et al. 1991; Honda et al.

1993; King et al. 1992; Tackley 1998; Kameyama and

Yuen 2006, 1998). Rapid progress of modern computing

technologies is providing us with effective means for

improving greatly the quality of spatial resolution and

reduction in the runtime of our numerical models.

Faced with the imminent threat of data tsunami pro-

duced by the ever-growing speed of multicore-computer

technology, we have now a dire need for a direct and

comprehensive method to display seamlessly the 3D spatial

and temporal characteristics of physical variables being

analyzed (Erlebacher et al. 2001; Rudolf et al. 2004;

Hansen and Johnson 2005, 2004), and the demand of

affordable and interactive visualization solution for the

large-scale numerical simulation datasets is inevitable in

geoscience community nowadays. In contrast to the tradi-

tional ways of visualization, such as the surface rendering,

isoline, isosurface, etc., the volume rendering technique is

becoming more and more attractive to geoscientists, who

are recently making significant inroads into parallel visu-

alization, using software tools such as PV3 and Paraview

(Jordan et al. 1996; Moder et al. 2007; Stegman et al.

2008). Previously, the multi-isosurface and streamline

representation methods have been preferred and they have

S. Chen (&) � H. Zhang � J. Zhang � Y. Shi

Laboratory of Computational Geodynamics,

Graduate University of Chinese Academy of Sciences,

19A Yuqanlu Ave., 100049 Beijing, China

e-mail: chenshi80@gmail.com

D. A. Yuen

Department of Geology and Geophysics,

Laboratory of Computing Science and Engineering,

University of Minnesota, Minneapolis, MN, USA

D. A. Yuen � S. Zhang

Minnesota Supercomputing Institute, University of Minnesota,

Minneapolis, MN 55455, USA

123

Vis Geosci (2008) 13:97–104

DOI 10.1007/s10069-008-0012-0

served to visualize successfully the 3D spherical mantle

convection (Wang et al. 2007; Moder et al. 2007). These

isosurfaces and streamlines can represent volumetric data

field(s) on a 3D surface by interpolating point data on

regular or irregular grids. The disadvantage lies in that it

lacks of showing successive scalar variations of physical

field accurately in 3D space. It is not enough to create one

beautiful image from the data. One needs to create hun-

dreds to thousands of such images at a time to make a

movie of the simulated mantle convection. Under these

circumstances, volume rendering technique has been

espoused in recent years for making movies. But 3D

volume rendering in the past was costly. For this reason,

this procedure was not popular, say 6 years ago. Volume

rendering does not use intermediate geometrical represen-

tations as usual. Rather it makes a sharp decision for every

voxel on whether or not one isosurface passes through it

and produces false positives (spurious surfaces) or false

negatives (erroneous holes in surfaces) simultaneously.

Particularly, this is very helpful in the presence of small or

poorly defined features of a physical field by offering a

distinct possibility for displaying weak or fuzzy surfaces,

which may be missed otherwise. This strategy allows for

substantial information to be displayed as a whole and

enables the researchers to discover new physical phenom-

ena hidden behind these preordained isosurfaces, which is

strictly enforced in surface rendering.

In the past few years volume rendering of regular grids

and texture-based techniques is becoming more mature and

easier to deploy. However, it is still very difficult to

implement effective algorithms of volume rendering on

unstructured grids or meshes from finite element method

(FEM) or finite volume method. The main difficulties lie

on how to obtain accurate order visibility for each cell. One

3D interpolation algorithm was proposed in this paper to

fulfill the transform from the unstructured grids (meshes) to

uniform grids. However, this resampling process is com-

putationally intensive and also results in numerical

artifacts. Therefore it still poses a very big obstacle for

efficient volumetric visualization, which is a first step

toward interactive visualization (Damon et al. 2007).

To solve these problems from volume rendering of

irregular grids and visibility ordering, Shirley and Tuch-

man (1990) presented the projected tetrahedra (PT)

algorithm for the rendering tetrahedral cells. Williams

(1992) also proposed the meshed polyhedra visibility

ordering (MPVO) algorithm for getting the visibility order

of cells before rendering them. In theory, the efficiency of

MPVO algorithm is not satisfying and leads to longer

rendering overhead in the runtime. We have investigated

these novel volume rendering techniques and found that the

hardware-assisted visibility sorting technique (HAVS)

(Callahan et al. 2005a, b) was suitable for implementing

visualization of mantle convection results over large

unstructured meshes in spherical geometry.

Our main objectives of this paper are on how to

implement the visualization of 3D mantle convection on

unstructured tetrahedral meshes and to present our expe-

rience in using these techniques in the Laboratory for

Computational Science and Engineering (LCSE) in the

University of Minnesota. Our paradigm of numerical

mantle convection simulation involves three main numer-

ical methods, as shown in Fig. 1. Our computation was

conducted at the BladeCenter of Minnesota Supercomputer

Institute (MSI). The visualized technique employed the

HAVS for volume rendering (Callahan et al. 2005). Current

version of HAVS uses the tetrahedron as the base cell for

the volume rendering of unstructured meshes. The source

code and compiled programs are freely available on the

Internet (http://havs.sourceforge.net).

Fig. 1 Overview of the visualization of 3D mantle convection with

unstructured meshes

98 Vis Geosci (2008) 13:97–104

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Numerical methods used in mantle convection

Realistic mantle convection is a strongly coupled nonlinear

system, which can be described by three governing non-

linear partial differential equations, the conservation laws

of mass, momentum, and energy (McKenzie et al. 1974).

The effects of Earth mantle’s heterogeneously distributed

rheology can be characterized with its strong temperature,

pressure and stress dependence (Zhong and Gurnis 1994),

the dynamical effects of phase transitions (Schubert et al.

1975) and multi-component chemical flow (Zhong 2006).

Different numerical approaches with unstructured meshes

have been utilized to address these problems. The

unstructured meshes can be employed to express the

majority of the physical phenomena. The transformation

between the different kinds of meshes is more easy and

affordable than the computational time spent in doing the

3D spatial interpolation.

Spectral method

The spectral method deserves great merit because it has a

higher accuracy than other numerical methods for the same

number of grid points, which are prescribed in the global

coordinate system. Zhang and Christensen (1993) have

developed a FORTRAN code for simulating 3-D spherical

mantle convection with lateral variations of viscosity due

to either temperature dependence and/or non-Newtonian

rheology. Higher-order finite difference method (Fornberg

1995) was used in discretizing the radial variations of field

variables (Zhang and Yuen 1995). The spherical harmonic

functions, which represent the latitude and longitude, are

used for approximating their lateral variations.The exis-

tence of lateral viscosity variations constitutes the

nonlinear mode coupling among the spherical harmonic

coefficients, which is treated by iterative numerical

approach. It has solved the problem of lateral variable

viscosity contrast up to 200 (Zhang and Yuen 1996).

We use this code to generate the temperature field over

the natural spherical coordinates. The generated data size is

257 along the longitude, 257 along the latitude and 34 in

the radial direction. Certainly this model is not a big one in

terms of the grid size. But it has been a challenging task to

visualize temporal flow motions using the traditional vol-

ume rendering technique, which requires the uniform

distribution of grids in 3-D Cartesian coordinates, cast in

global coordinate system. Mapping the grids in spherical

coordinates to the Cartesian geometry requires numerical

interpolation, which decreases the accuracy. On the other

hand, extremely dense grids are needed to get rid of the

fake circles appeared on the core mantle boundary. Now

the HAVS technique seems promising to allow us to tackle

the visualization difficulty. For using HAVS method, first

we need to construct a tetrahedron mesh, which can be

implemented in the code directly for the output. More

details about how to employ the HAVS technique are

presented in ‘‘Strategy of volume rendering with unstruc-

tured mesh’’.

Finite volume method with ACuTEman

The finite volume method is a powerful numerical method

for solving differential equations with strong nonlinearity

and hyperbolic characteristics (Pantakar 1980). It also

shares common merits with finite element and finite dif-

ference methods. According to this method, the discrete

equations, second-order correct, are obtained by integrating

equations over a control volume or cell. In the field of 3D

mantle convection numerical simulation, Kameyama

(2005a), Kameyama and Yuen (2005b) has devised a new

simulation code of mantle convection in a 3D spherical

shell named as ACuTEMan. The major innovation of the

code is that it has used two techniques as Yin-Yang Grid

(Kageyama et al. 2005c) and ACuTEMan (Kameyama and

Yuen 2006) algorithm. The Yin-Yang grid is an effective

spatial discretization procedure which involves overlapping

of two grids on spherical geometry whose boundaries

resemble the seams of a baseball, shown in Fig. 2. This

method overcomes the problem of the polar singularity

inherent in spherical harmonic expansion. The ACuTEman

algorithm, originally written for a Cartesian geometry, has

recently been redesigned for large-scale spherical mantle

convection problems.

In this section, we present a feasible approach for vol-

ume rendering, using the data generated from the

ACuTEman (Kameyama 2005a) program. This program is

different from the previous method in terms of the Yin-

Yang grid processing, being used here. We need to com-

bine the first two parts of Yin-Yang grid as one volumetric

dataset and then re-mesh this to tetrahedral cells.

To achieve this goal, first we have employed the Amira

software to pre-process the dataset format. Amira is a

Fig. 2 The Yin-Yang mesh scheme for spherical geometry

Vis Geosci (2008) 13:97–104 99

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comprehensive commercial tool for Geosciences Visuali-

zation. It was developed by Mercury Computer Systems,

and available for Windows, Uni, Linux, and Macintosh

(http://www.tgs.com) platforms. There are hundreds of

modules within the entire software packages. In our case,

we only used the LatToHex and HexToTet modules to re-

mesh and apply a hybrid module for assembling together

the Yin-Yang grid (Fig. 3a). After assembling the Yin-

Yang grid, we then used GridEditor tool for optimized the

intersection of re-mesh cells (shown as Fig. 3b). We need

to carry out a smoothing process for visualization in the

part of intersection. Figure 3c, d show the results of

the combined grid using data smoothing process. Finally,

the structured meshes in spherical system are transformed

to the tetrahedral meshes.

Finite element method with Citcoms

The FEM is very effective in solving differential equations

with complicated geometry and variable material proper-

ties, in this vein we employ the CitcomS (Moresi and

Solomatov 1995), which was developed by Moresi and

Solomatov (1995), to simulate mantle convection process

in this section. Citcoms is a robust finite element package

with the capability of solving thermal convection problem

within a spherical shell. In the specifications of Citcoms,

many parameters can be defined for simulating realistic

mantle convection problem. Figure 4a is one cap for

regional convection computation and Fig. 4b shows how to

combine the total 12 caps into a single spherical convection

model.

The datasets used in this paper came from the incom-

pressible mantle convection model with Stokes equation

and temperature-dependent viscosity. The Rayleigh num-

ber is 5 9 108, and the full model consists of 12 caps for

parallel computing. The resolution of each cap is

17 9 17 9 17 points. The visualization results show as the

Fig. 5 using volume rendering and isosurface techniques.

For each cap, we need to generate the tetrahedral cells

using the Amira modules. The process is similar to the

methods described in the previous section (shown as in

Fig. 3). Because there is no intersection among the 12 caps,

the pre-processing can be fulfilled easily by a method

consisting of a combination of several directed steps

(Fig. 5).

Fig. 3 Pre-processing of the

Yin-yang mesh

Fig. 4 The one cap and full model scheme in Citcoms (Tan et al.

2007)

100 Vis Geosci (2008) 13:97–104

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Strategy of volume rendering with unstructured mesh

Volume rendering technique is widely used to display the

3D discrete scalar fields. The optical model equation has

been used to create an image. Image processing is used to

integrate the individual contribution from the object space

to the image space along the viewing ray direction, using

front-to-back or back-to-front compositing algorithms.

Direct volume rendering technique is more effective than

the traditional isosurface operation for visualization of a

scalar field. However, this method requires each sample

value to be mapped to opacity with a certain color scale.

This technique is accomplished with a ‘‘transfer function’’,

which can be a simple ramp, a piecewise linear function or

an arbitrary table. Once converted to a RGBA (for red,

green, blue, alpha) value, the composed RGBA result is

projected with correspondence of the frame buffer pixels.

However, for unstructured gird, the sample value and

composed RGBA result and pixel projection are consider-

ably complicated. The PT Algorithm (Shirley and Tuchman

1990) was the first solution for rendering tetrahedral cells

using the traditional method. It appears that, the imaging

speed will depend on the algorithm validity. With modern

GPUs, Farias et al. (2000) has presented a new volume

rendering algorithm for projection of unstructured volume

rendering. Based on Farias’s work, Callahan et al. (2005a, b;

2006) developed a new technique that uses rendering cal-

culation and distributes part of this task to CPU, and then uses

the GPU to complete the entire task. We will now put forward

a new concept, which we will call ‘‘k-buffer’’ (Fig. 6).

Fig. 5 Volume rendering of 3D mantle convection generating from

the Citcoms FEM software. Here the local coordinate system is

employed for each spherical sector

Fig. 6 The HAVS solution for unstructured mesh (Callahan et al.

2005)

Fig. 7 Flowchart of volume

rendering

Vis Geosci (2008) 13:97–104 101

123

In this section, we present a simple flowchart (Fig. 7) to

show the implementation of visualizing of mantle convec-

tion simulation results. The entire process employed the

Amira software for pre-processing and constructed the

lookup table, afterwards the volume rendering was finished

by the HAVS technique. For the different datasets generated

by the various programs, we only need to consider how to

rapidly obtain the requirement cells for HAVS algorithm.

The transfer function can be conveniently acquired by using

the colormap editor of provided by Amira modules, which

was fully discussed in ‘‘Transfer function designed for

specific features of mantle convection’’.

Based on the above discussion, and after some practice,

we gained deep experience on how to utilize HAVS tech-

nique for making interactive visualization of mantle

convection. The rendering performance of HAVS is

acceptable for our large-scale dataset generated by the

three software packages for the mantle convection prob-

lems. The time of rendering with 10 million tetrahedral

cells is about 5–6 s on our workstation (SUSE Linux, AMD

64 bits duo core, Nvidia Quadro 256 M graphic card).

Figure 8 shows that it appears difficult to find some arti-

facts in the rendering result. For future research in

interactive visualization, we will be able to change easily

the LUT setting to explore the interesting features for the

dataset of mantle convection. The time taken of the entire

process is not so long.

Transfer function designed for specific features

of mantle convection

The transfer function describes the mapping from grid

values to renderable optical properties, such as the opacity,

color field, emittance, etc. An evident characteristic of the

transfer function for volume rendering can be described as

a time-consuming and not intuitive task. The design of

effective color and opacity transfer functions from scalar

values has been the subject of substantial research over the

past decade (Fang et al. 1998; Fujishiro et al. 1999). For the

mantle convection problems, our main objective is to find

the specific features of the convection pattern, such as the

vertical boundary layers of plumes and slabs.

In general, the audience looks at the upwelling, down-

welling structures and detailed local features such as the

eruption of boundary layer instabilities. Therefore, the

transfer functions with different characteristics are required

to direct volume rendering, and it can be either generated

automatically or edited interactively using an intuitive

color-map editor. In this paper, we employed Amira to

create a transfer function.

Using the color-map editor of Amira, we can semi-

automatically edit the opacity and RGB color associated

with the histogram of the original data values. Based on the

instructions of Amira for the ‘‘am’’ file format of color-

map, we can obtain the result of lookup table and convert it

to input HAVS program consequently.

Figure 9 shows a snapshot of the upwelling pattern

associated with the colormap setting of Amira. It presented

the illustrative specific features of mantle convection.

Using the appropriate opacity strategy, we can explore the

upwelling characteristics easily. Figure 10 is another color-

map setting for the same dataset. In these two figures, the

black line of left figure expresses the opacity, the red line,

green line and blue line express ‘‘R’’, ‘‘G’’ and ‘‘B’’ colors.

From the comparison between these two black lines, we

find that the opacity of the low value region in Fig. 10 is

wider than that of Fig. 9, the opacity of the high value

region in the Fig. 10 has been entirely suppressed. Thus

far, the transfer function can help us explore the specific

convection features much more effectively.

Discussion and perspectives

We have discussed some of the problems arising from

implementing volume rendering with unstructured mesh in

spherical geometry, using the HAVS technique. Three

approaches of numerical simulation of mantle convection

have been applied for the volumetric visualization tests.

Our experience has shown that the technique of HAVS is

Fig. 8 Volume rendering of the 3D mantle convection of spherical

harmonic based the HAVS technique with 10 millions of unstructured

mesh

102 Vis Geosci (2008) 13:97–104

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feasible for visualization of the 3D datasets generated by

these numerical simulation codes. HAVS offers a great

approach for volume rendering of scalar field that cannot

be easily done in the traditional way. However, we stress

here that this is still at a nascent stage, or has evidently

some drawbacks. The lack of the capability of interactively

adjusting color-map and opacity results in the inefficient

use of the package. Compared the traditional volume ren-

dering, HAVS performs better. But the entire visualization

process needs to be accelerated in order to make this tool

more usable and attractive to the potential user. HAVS is

an open-source package. People interested in volume ren-

dering can get into the code and make changes or add more

features. In coming work, more effective methods should

be explored and be implemented in the near future, such as

the capability to edit the color-map automatically by using

ambiguous or specified rules, and this improvement can

lead to expansion of the compatible file formats on the

basis of HAVS technique. By doing this we can take a step

in the right direction of visualization in 3D spherical

mantle convection. Such a tack will bode well for us in the

future, especially in light of recent proliferation of GPU

cards. More research work is needed to improve the ren-

dering speed by better using the rapidly improved GPU or

the CELL architecture and their respective programming

models.

Acknowledgments The authors would like to thank Mr. Yunhai C.

Wang and Dr. David Henry Porter from M.S.I. for constructive dis-

cussions. We acknowledge that Dr. Masanori C. Kameyama has

kindly provided us with his AcuteMan code. We thank Ms. Stephanie

Chen for technical assistance. This project is jointly supported by

National Basic Research Program of China (2004cb408406) and

National Science Foundation of China under grants number

(40774049, 40474038). The parallel simulation program is supported

by Supercomputing Center of Chinese Academy of Sciences

(INF105-SCE-02-12). Dr. David A. Yuen thanked NSF for support in

CMG and ITR programs.

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