精确追踪变形材料表面的拓扑变化方法

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"Deforming Meshes that Split and Merge" 这篇SIG09的文章主要探讨了一种针对变形材料表面跟踪的创新方法，该方法能够有效地处理拓扑变化，如网格的分裂和合并。作者采用拉格朗日表面跟踪技术，利用三角网格作为表面表示，以保留精细的特征。在特定网格分辨率下，当检测到融合或分离事件时，他们会在受影响的单元格中局部创建新的网格部分，使用标准等值面创建方法。新生成的、拓扑简化后的网格部分会与原有网格在单元边界处缝合。这种方法强调在保持细节特征的同时，对非可见部分进行简化。该表面追踪算法不依赖于特定的可变形材料模拟方法，展示了与两种截然不同的模拟器配合的效果：一种是使用四面体元素的拉格朗日有限元法（FEM）模拟器，另一种是欧拉网格基流体模拟器。尽管该跟踪方法具有通用性，但特别适用于显示精细表面细节和大量拓扑事件的模拟。研究亮点包括粘弹材料的合并（如复杂几何形状的物体碰撞），具有许多折叠和合并事件的拉糖动画，以及坚硬塑料材料的拉伸和切割。图1展示了在欧拉流体模拟中落下粘弹性球的例子，即使经过空气移动和地面撞击，可见表面仍能保持其细节。这篇文章的核心知识点包括： 1. 拉格朗日表面跟踪技术：这是一种追踪变形物体表面的方法，通过跟踪物体上的固定点来模拟表面的变化。 2. 三角网格表示：使用三角形作为基本元素构建网格，可以精确地表示物体的细节特征。 3. 拓扑变化处理：在检测到网格的融合或分裂事件时，局部生成新的网格结构，并在单元边界处进行缝合，以适应拓扑变化。 4. 标准等值面创建方法：用于生成新的网格部分，确保拓扑结构的正确性。 5. 通用性与适应性：该方法能与不同类型的物理模拟器结合，如拉格朗日FEM和欧拉流体模拟，适用于各种变形场景。 6. 细节特征的保护：在进行拓扑简化时，着重保护表面的精细特征，同时简化不可见部分，以优化计算效率。 7. 应用示例：包括粘弹材料的合并、复杂的拉糖动画和塑料材料的力学行为模拟，这些都展示了方法在处理复杂拓扑事件和保持表面细节方面的有效性。通过上述技术，研究人员和动画师能够在模拟过程中更好地捕捉和表现物体变形的动态特性，特别是在涉及拓扑变化的复杂场景中，提供更加真实和视觉上吸引人的效果。

Deforming Meshes that Split and Merge

Chris Wojtan Nils Th

urey Markus Gross Greg Turk

Georgia Institute of Technology ETH Z

urich ETH Z

urich Georgia Institute of Technology

Figure 1: Dropping viscoelastic balls in an Eulerian ﬂuid simulation. Invisible geometry is quickly deleted, while the visible surfaces retain

their details even after translating through the air and splashing on the ground.

Abstract

We present a method for accurately tracking the moving surface of

deformable materials in a manner that gracefully handles topologi-

cal changes. We employ a Lagrangian surface tracking method, and

we use a triangle mesh for our surface representation so that ﬁne

features can be retained. We make topological changes to the mesh

by ﬁrst identifying merging or splitting events at a particular grid

resolution, and then locally creating new pieces of the mesh in the

affected cells using a standard isosurface creation method. We stitch

the new, topologically simpliﬁed portion of the mesh to the rest of

the mesh at the cell boundaries. Our method detects and treats topo-

logical events with an emphasis on the preservation of detailed fea-

tures, while simultaneously simplifying those portions of the ma-

terial that are not visible. Our surface tracker is not tied to a par-

ticular method for simulating deformable materials. In particular,

we show results from two signiﬁcantly different simulators: a La-

grangian FEM simulator with tetrahedral elements, and an Eulerian

grid-based ﬂuid simulator. Although our surface tracking method is

generic, it is particularly well-suited for simulations that exhibit ﬁne

surface details and numerous topological events. Highlights of our

results include merging of viscoplastic materials with complex ge-

ometry, a taffy-pulling animation with many fold and merge events,

and stretching and slicing of stiff plastic material.

Keywords: Physically Based Animation, Deformable Meshes,

Fluid Simulation, Topological Control

1 Introduction

The physical world is rich in materials that deform. In many in-

stances, these materials easily undergo topological changes, allow-

ing separation or joining of components. Such materials include

water, toothpaste, bread dough, peanut butter, taffy, tar, and clay. In

order to produce plausible animations of these deforming materials,

we must develop simulation techniques that retain visual details, but

at the same time allow topological changes to the surface. This is

the focus of our work.

We introduce a method of tracking and updating the surface of

a deforming object in a manner that gracefully allows for topology

changes. Our approach uses a mesh to represent the surface of ob-

jects. The key beneﬁt of using a mesh is that it allows the surface to

be moved in a Lagrangian manner, using a high-order ODE integra-

tor to move the vertices of the surface. This results in the retention

of ﬁne surface details. The drawback of using a mesh surface is that

changing the topology of a mesh is notoriously tricky, both in terms

of robustness and program complexity. Our approach avoids these

difﬁculties by replacing those portions of the mesh where a topol-

ogy change should occur with a simpliﬁed surface that is generated

using a standard isosurface creation method. This new portion of

the mesh is then stitched together with the untouched parts of the

surface.

Our approach to topology change lets us decide which potential

topological modiﬁcations are permitted and which ones should be

prevented. This allows us to customize the behavior of the simu-

lator in order to simulate materials with particular attributes. For

example, we can allow surface separation events, but forbid surface

merging. This might be appropriate for simulating bread dough that

has been rolled in ﬂour so that it no longer sticks to itself. One par-

ticular rule for marking where a merge event should be triggered

is what we refer to as the deep cell test. This test allows surface

merging when it is deeply embedded in the folds of a material, but

it leaves alone potential merge events that are near the surface and

that might produce visually distracting results.

The key attributes of our approach to surface tracking are as fol-

lows:

• Our algorithm performs topological splits and merges, which

decreases the memory and computation requirements of deal-

ing with complicated surfaces and permits interesting behav-

iors like the merging of water droplets.

• Our simulator retains thin features and ﬁne surface details.

• The method is robust, even during complex topology changes.

• The surface tracker can be easily tuned to allow or forbid par-

ticular kinds of topology changes to better suit a given mate-

rial.

• Our surface tracker does not depend upon any particular sim-

ulation technique. In particular, we demonstrate results from

both an Eulerian ﬂuid solver and an FEM simulator.

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