Accurate Real-time Complex Cutting in Finite Element Modeling

Tong Xin, Pieran Marris, Ana Mihut, Gary Ushaw and Graham Morgan

School of Computing, Newcastle University, Urban Sciences Building, 1 Science Square, Science Central,

Newcastle upon Tyne, U.K.

Keywords:

Cloth Simulation, Cutting, FEM, Real Time.

Abstract:

This paper presents a real-time method for enacting accurate cutting of thin materials while retaining the

physical accuracy of the underlying deformable material model. Our primary contribution is to offer a ﬂexible

real-time solution that can allow accurate cuts, re-cuts, curved cuts, sectioning (cutting out) on a deformable

FEM model. Further contributions include improved handling of mesh element resizing after cutting for

increasing material stability while balancing such resizing against real-time requirements. In this manner, we

can represent precise cut incision (which may be curved or irregular) by subdividing mesh elements locally (at

the cut point) while negating inappropriate (ill-shaped) elements with real-time level computational overhead

and so avoiding modeling instabilities efﬁciently. We present the results of our solution that show the accuracy

of the cutting method and illustrate timings for our optimised approach demonstrating its real-time qualities.

1 INTRODUCTION

As real-time simulations have provided models rep-

resenting increased visual realism, matching this with

physical interaction realism is required. One such in-

teraction property that has received signiﬁcant inter-

est is that of cutting; acting cuts into simulated ar-

tifacts in real-time and having the results persist. It

has been explored more in those applied areas of re-

search that favor accuracy of precision and material

behavior (e.g., simulated surgery) and less so in those

favors aesthetics over interaction (e.g., video games).

It brings about a balance between physical proper-

ties and aesthetic appearance in cutting. To achieve

precision and accuracy while maintaining appropriate

(believable) interaction requires a material model of

advanced features that impact negatively on overall

real-time performance.

The ﬁnite element method (FEM) is a com-

mon choice for constructing accurate models of de-

formable material behavior. The model has properties

that allow accurate inspection of stress and other re-

lated parameters to determine the potential real-world

behavior. FEM is computationally more expensive

to achieve than its counterparts that afford visually

pleasing simulations at the expense of physical ac-

curacy (e.g., mass-spring). However, to achieve re-

alism in cutting and leave a post-cut model that be-

haves as expected in the real world then FEM is a

clear choice. However, difﬁculty arises in reorganiz-

ing the mesh of the FEM (the representation of the

surface area) to ensure accurate cut depiction (the cut

is where one desires) and resultant models still exhibit

appropriate physical properties (no irregular behavior

is introduced due to cuts). It requires a suitable algo-

rithmic solution to ensure the reorganization, and pos-

sibly introduction or removal, of the elements (usu-

ally triangles) that constitute the mesh representing

the model under dissection. Presenting algorithmic

solutions that are engineered to work in real-time is

the main focus of this paper.

Achieving accurate cut incisions does require el-

ements to be altered or introduced. If a cut is to be

presented exactly where a user needs, then elements

must be reconstituted around this area, usually, it re-

quires introduce additional mesh elements. However,

achieving accurate cuts while making appropriately

shaped elements is a non-trivial problem at the heart

of a real-time FEM cutting technique. Ill-shaped ele-

ments relate to the inner working of the FEM; each el-

ement represents a unit of deformability that is solved

to present an overall ”shape” of the material model.

As discretization is required as a step to attaining a

practical solution, ill-shaped objects can inject signif-

icant errors in the cumulative mathematical calcula-

tions. It is compounded in real-time solutions where

equation solvers accomplish a solution within a time

boundary. A time boundary too low would provide in-

sufﬁcient time to achieve the desired mathematical ac-

curacy that, ultimately, will cause the material model

Xin, T., Marris, P., Mihut, A., Ushaw, G. and Morgan, G.

Accurate Real-time Complex Cutting in Finite Element Modeling.

DOI: 10.5220/0006542801830190

In Proceedings of the 13th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2018) - Volume 1: GRAPP, pages

183-190

ISBN: 978-989-758-287-5

183

to fail.

The more complicated and ﬂexible a material

model is regarding cutting the more compounded the

difﬁculty of ensuring real-time and physical accu-

racy. That is certainly true when allowing re-cutting

as already altered and newly introduced mesh ele-

ments must be re-altered and managed while main-

taining real-time behavioral properties. As mesh ele-

ments growing (more cuttings) so the computational

expense increases and the possibility of ill-shaped ele-

ments also increases; this is when existing approaches

tend to have the greatest difﬁculty, and many do not

attempt re-cutting, especially in real-time scenarios.

In this paper, we continue the line of research

into the cutting of material models using FEM. How-

ever, we present a general solution applicable to ad-

vanced material models that can be simulated in real-

time. We present solutions to re-meshing of the mate-

rial model after cutting that minimizes ill-shaped el-

ements while accurately reﬂecting the exact position

of the desired cut. Our novel approach provides con-

vincing cuts that can be straight or curved, appear ex-

actly where directed, allows re-cutting and the abil-

ity to cut-out (resulting in multiple FEMs) and do so

while maintaining the physical properties of the cloth

model itself.

2 BACKGROUND AND RELATED

WORKS

Our primary focus is to generate aesthetically pleas-

ing cuts to dynamic, responsive, material models

while retaining accurate physics behavior in real-

time. The cutting of deformable material has been

an active area of research for many years (Wu et al.,

2015). In this section we provide a broad discus-

sion of techniques that use a variety of models, how-

ever, for brevity we only afford additional debate for

those works that are focussed on (but not necessarily

achieved together): (1) accurate cutting; (2) FEM; (3)

real-time; (4) re-meshing.

2.1 Cutting

In mesh-based cutting techniques, non-progressive

approaches have advantages by which computational

overhead can be minimized using mesh subdivision

determined as a whole, and present a cut at any de-

sired point (Pietroni et al., 2009; Steinemann et al.,

2006). An alternative approach is to use XFEM,

where the cut can be described separately as a func-

tion on top of the higher order model (Kaufmann

et al., 2009). The latest work in this area is com-

pelling, allowing cutting of 3D objects (Koschier

et al., 2017). The placement of the cut can be very ac-

curate. However, the cut edge can hardly inherit fun-

damental physics with the approximate enrichment

basis functions.

Besides the persistence of cuts, the ability to re-cut

areas must be considered carefully. As the topology

of the mesh could be partially split numerous times

before the current cut, care must be taken to assure

the subdivision of the start and end elements is prop-

erly handled. The meshless model approach proposed

by (Nesme et al., 2009) and polygon model (Sifakis

et al., 2007) present abilities to re-cut, but such mod-

els are unlikely to ﬁnd favor in real-time simulations

that leave the underlying mesh altered.

2.2 Topology

Due to its relatively simple construction and con-

vincing behavior the spring-based system (Baraff and

Witkin, 1998) has been widely used for material

modeling cutting (Cotin et al., 2000; Souza et al.,

2014). Unfortunately, the model relies on the the

springs connectivity and once the pattern is altered,

behavior is not convincing. FEM employee dynam-

ically reconstructs each of the element interactions

based on the shape function, in this manner force

is enacted across elements rather than independently

across springs (Cakir et al., 2009). FEM then is ex-

ploited to provide more physically accurate deforma-

tions (Yeung et al., 2016).

Nonlinear FEM approaches employee nonlinear

solver to handle deformation, while they are computa-

tionally more expensive than linear FEM. This can be

largely mitigated with introducing co-rotational FEM

(Georgii and Westermann, 2008). It adopts linear

stress/strain solver on each element, and pre-compute

a single stiffness matrix which is then simply trans-

formed into the current conﬁguration. It provides a

fast FEM simulation with only a negligible overhead

compared to mass-spring systems. Therefore it has

been used in cutting simulations (Courtecuisse et al.,

2010; Turkiyyah et al., 2011).

2.3 Subdivision

Subdivision algorithm provides an opportunity for re-

cutting the same area on material models. Edge split-

ting changes the mesh connectivity (Souza et al.,

2014; Yeung et al., 2016), whereas other methods

such as vertex-snapping merely move the nodes to

along cut path (Serby et al., 2001). Although it pro-

vides pleasing results, on a coarse mesh it can often

GRAPP 2018 - International Conference on Computer Graphics Theory and Applications

184

lead to jarring effects. For this reason, we handle edge

splitting using current dynamic subdivision (Bielser

et al., 2004) with the non-uniform subdivision. How-

ever, other edge splitting methods often result in for-

matting ill-shaped elements.

Adaptive subdivision schemes verify the level of

subdivision depending on the force intensity acting on

the mesh so that can produce a high-quality mesh in

physics-based deformation (Bender and Deul, 2013;

Narain et al., 2013; Koh et al., 2014), tearing (Pfaff

et al., 2014) and force-based cutting (Seiler et al.,

2011). These methods enforcing uniform subdivision

avoiding ill-shaped elements. However, this also in-

curs a substantial computational overhead and thus in

many cases a coarse subdivision level is deﬁned.

The Delaunay approach has been used (Nienhuys

and van der Stappen, 2004; Busaryev et al., 2013)

to limiting ill-shaped elements. A further issue is

the problem of computational complexity, although

vastly optimized, Delaunay triangulation is still de-

pendent on the problem size and can be computation-

ally expensive.

3 THEORETICAL BACKGROUND

Mesh quality is a crucial factor in FEM-based cutting

techniques for maintaining numerical stability and ro-

bustness of the simulation. The Delaunay reﬁnement

scheme is a well-established technique for maintain-

ing mesh quality (Bowyer, 1981). It is achieved by

ensuring that no nodes are created within the circum-

circle of existing triangles, as illustrated in Figure 1.

The effect of this is to ensure that the minimum angle

of created triangles is maximized, thereby improving

mesh quality.

The Bowyer-Watson algorithm reduces computa-

tion cost by eliminating the need for complex topo-

logical validity checks (Rebay, 1993). The approach

incrementally inserts one node at a time, and then lo-

cally validates the triangulation of a subset of the de-

sired points. It can take O(N × log N) operations to

triangulate N points. With the consideration of lim-

ited computation cost in real-time applications, we

adopt the Bowyer-Watson algorithm to generate ini-

tial Delaunay triangulation.

4 IMPLEMENTATION

Our approach combines a Delaunay-based reﬁnement

technique (Bowyer-Watson algoritm) on co-rotational

ﬁnite element modeling. It ensures that the higher

computational cost of a subdivided mesh is focused

on the areas of material closest to the simulated cut

while maintaining the quality of the modiﬁed mesh.

Furthermore, we apply a local split optimization pro-

cess to handle the ill-shaped triangles.

4.1 Mesh Cutting

When operating subdivision method as the number of

modiﬁcation increases, there is a necessary trade-off

of ﬁdelity of simulation and mesh quality. Conse-

quently, a technique which focuses on both the sub-

division of elements in the area of the mesh most af-

fected by the cut and the quality of generated mesh

elements can lead to an optimal solution.

4.1.1 Locality of the Cut

The ﬁrst step in simulating a cut to the material is to

identify the triangles of the mesh that are affected.

The process of Delaunay triangulation and incision

generation will take place in the local coordinate

space of the material mesh with no deformation due

to external forces. However, the causes such as col-

lisions takes place in world space, where the material

is in a deformed state.

For our simulation, a mouse is used to identify the

location of an incision in the world space. The po-

sition of the mouse in the 3D world space is trans-

lated to an un-deformed local space for the material

mesh. The path of the incision is generated from a

cubic Bezier curve which requires four control points

(four mouse clicks).

Intersection detection will operate between the cut

path and each edge of triangles of the mesh. Triangles

with one or two line intersections will then be consid-

ered for generating initial Delaunay triangulation in

the next stage of the algorithm. New points will be

inserted at the exact place where the cut crosses the

triangle edge.

4.1.2 Initial Delaunay Triangulation Generation

An example of traditional and Delaunay subdivision

is shown in Figure 1. In the case of the traditional

Figure 1: Delaunay generation principle.

subdivision, an ill-shaped triangle has been generated,

Accurate Real-time Complex Cutting in Finite Element Modeling

185

and there are nodes of other triangles within its cir-

cumcircle. The right diagram illustrates the Delaunay

triangulation in which no circumcircle of any triangle

in the triangulation includes nodes from other trian-

gles.

The two instances of triangle cut types are illus-

trated in Figure 2. A new node is created on any in-

Figure 2: An example triangle with two intersection points

(left), and an example triangle with one intersection point

(right).

tersected edges at the position of intersection (i.e. one

for each side of the cut at each intersection to open

the cut incision. The Bowyer-Watson algorithm then

regenerates the original triangle shape made up from

constituent triangles with the three original points and

new points. Consequently, the original triangle is sub-

divided into two or three constituent triangles. Espe-

cially, when a cut is terminate inside the triangle the

original triangle will be replaced by four small trian-

gles as shown on the right of ﬁgure 2, and the incision

will also terminate inside the triangle. In the case of

two intersection points, one of the triangles is sepa-

rated by duplicated nodes.

Any ill-shaped triangles generated in this step will

be recognized and grouped for the optimization oper-

ation.

To generate arbitrary incisions (e.g. intersected

cuts), the connectivity between the affected triangle

and its neighbors is considered when opening cut in-

cision. After recognizing an intersected edge, our

method identiﬁes whether there is another triangle

that shares that edge. In that case, the new point and

its duplicate are shared by the affected triangle and its

neighbor.

4.1.3 Localized Optimization

In the linear ﬁnite element method, the computa-

tional speed can be reduced by inefﬁcient elements

in the mesh. Consequently, optimal mesh generation

and modiﬁcation are required to produce high-quality

mesh elements. This issue has been speciﬁcally iden-

tiﬁed in triangular meshes (Shewchuk, 2002) where

irregular sizes or shapes of triangles can have a detri-

mental effect on computation time. While the Delau-

nay algorithm can maximize the minimum angle of

all triangles in the triangulation, this does not guar-

antee that no ill-shaped triangles will be generated.

To address this problem, we introduce local split opti-

mization to increase the quality of the modiﬁed mesh.

We identify the aspect ratio as the parameter to

efﬁciently evaluate our generated triangles. For trian-

gular meshes, aspect ratio is deﬁned as:

AR = l

max

(1)

where l

max

is the length of the maximal edge, and a

max

is the height measured from this edge. The approxi-

mate minimum possible aspect ratio for the original

triangles in the mesh is 2. In our local split opti-

mization algorithm, newly generated triangles whose

aspect ratio is more than 3 will be re-triangulated as

shown in the right of Figure 3. The steps of our opti-

Figure 3: An example of the localized optimization.

mization operation are:

• The two longest edges are selected ﬁrst;

• New points are inserted at the mid-point of the

selected edges;

• Any ill-shaped triangle (as deﬁned by aspect ratio)

is regenerated by the Delaunay algorithm;

The introduction of new points necessitates the

subdivision of the triangles that are directly adjacent

to the new points (referred to as the 1-ring triangles).

During the subdivision step, if an edge is subdivided,

but its neighbor is not, a discontinuity can be created

(as illustrated by the shaded section in Figure 4) in

the form of a crack. Avoiding this cracking of the

mesh, the neighboring 1-ring triangle is also subdi-

vided with the new points, replacing the triangle with

two constituent triangles (as shown in the right of the

ﬁgure). Note that only one of the 1-ring triangles re-

quires this subdivision (rather than all of them as in

the more generalized subdivision methods). Further-

more, if there is any ill-shaped triangle created during

this step, our optimization process will keep operating

until the triangulation is under the acceptable level.

4.2 Topological Modiﬁcation

Employing a linear elasticity model with an implicit

integration scheme allows us to generate a fast and

GRAPP 2018 - International Conference on Computer Graphics Theory and Applications

186

Figure 4: A crack appears when its neighboring triangle has

been subdivided (a), 1-ring neighbors of subdivided trian-

gles are split as well (b).

stable simulation. However the approach can also re-

sult in unrealistic or unstable effects when large defor-

mations are needed. To resolve this issue, we use co-

rotated ﬁnite element modeling (Felippa and Haugen,

2005; Georgii and Westermann, 2008), which extract

the non-linear part of the current deformation before

computing the forces.

As cutting of a mesh necessitates disconnecting

parts of that mesh, a topological modiﬁcation is re-

quired. Changing any element of the global matri-

ces invalidates any previous factorization, necessitat-

ing a refactorization of the matrices. Rather than re-

constructing the global matrix in each time step, we

incrementally update the global matrix from the cur-

rent state in two steps: index replacement and element

addition.

4.2.1 Index Replacement

In contrast to current methods, we change the vertex

indices of all intersecting faces so that the contribu-

tions of the original elements are not removed but re-

placed in the global stiffness matrix K. For instance,

if in time (i) the cut path passes through the element

two vertex indices of the element are replaced by

the indices of new points to ﬁll in one side of the in-

cision. In the next step (i + 1) the element stiffness

matrix K

must be updated. The initial area A

will

be calculated from the new vertex positions. As dis-

crepancies in vertex indices can lead to computational

inefﬁciency, we mitigate this effect by changing the

vertex indices in this step.

4.2.2 Element Addition

After the index replacement step, new vertices are cre-

ated to prepare for element modiﬁcations, so the di-

mension of matrices will be increased to accommo-

date the new vertices. Rather than reconstructing the

global matrix in each time step, we incrementally up-

date the global matrices. For instance, the stiffness

matrix K is updated to:

= K

∑

i=1

(2)

where K

and K

represent the current stiffness matrix

in this time step i and the updated stiffness matrix in

next time step i + 1, m is the number of new elements,

and G

are the matrices which map the rows

and the columns of the new element metrics to the

global matrix. The mass and damping matrices must

also be updated in an equivalent way.

The new vertices introduced in the subdivision of

the cut triangle, and the affected neighboring trian-

gles, are each added to the global matrices in this

manner. This dynamic approach, incorporating each

new vertex into the index list incrementally (in two

steps as described), is more computationally efﬁcient

than recalculating the global matrix for the entire

mesh.

4.3 Time Integration

We employ an implicit time-based solver to simulate

the evolution of the visco-elastic deformable surface

model.

= Kx + Kx +

Kx + f

(v)

+ f

ext

(3)

Where x is a matrix containing each of the n vertices

in the model. For the time step t when the cutting

path is instigated, computation is completed by the

following steps:

• Compute the rotations R

for each triangle.

• Update the stiffness matrix K with the two steps

described in section 4.2, assemble the global ma-

trix

K (rotate the result of multiplication back to

3D space), and compute the viscous bend force

(v)

with velocity v.

• Compute Kx, where K and x stand for the stiffness

matrix and the vector of position in the rest state

respectively.

• Calculate the total forces f

ext

which include grav-

ity and any other external force.

• Carry out an implicit time step.

The key of this procedure is that the system of equa-

tions to be solved in each time step is linear. The

linear system is been solved by a conjugate gradi-

ent method. Our topological modiﬁcation method de-

creases the computation involved in recomputing the

matrix for every node, as we incrementally add nodes

through re-indexing.

5 RESULTS AND DISCUSSION

In this section, we discuss the performance of our

approach in a variety of contexts, including visual

Accurate Real-time Complex Cutting in Finite Element Modeling

187

outputs for complex cuts, computation time, mesh

quality and level of detail of cut surface. The

algorithm was tested on a series of meshes ranging

from 800 triangles to 12800 triangles, and for each

resolution the simulation operation was repeated 10

times to obtain the average computation time. All

experiments were run on a desktop PC, equipped

with an Intel



Core

T M

i7 − 4790S CPU @ 3.20GHz

processor (we use a single core), 16GB of RAM ,

and NV IDIA GeForce GT X 980 graphics card. Dy-

namic simulations of the proposed implementation

can be found in the accompanying video linked to

https://youtu.be/Yug16Wi1GgA.

5.1 Complex Cut Surfaces

Our method can produce realistic and accurate cut

surfaces with arbitrary shapes even in low-resolution

meshes. Note that all meshes illustrated in ﬁgures are

constructed by 800 initial triangles. Figure 5 shows

two example cuts (straight and curved). Figure 6 illus-

trates partial cuts which include cut from an edge (the

left) and cut part of mesh entirely out (the right). Fig-

ure 7 shows a re-cut (the left) and multiple cuts (the

right). Within a single mesh. Intuitively, our method

can provide a high-detailed cut surface independent

of resolution requirements. Moreover, the visual out-

puts also demonstrate how our method can produce

realistic cut incision with accurate physical behaviors

which involve substantial bending, swirling and ﬂip-

ping.

Figure 5: Different shapes of single cut within a mesh. The

left shows a straight cut, while the right shows a curved cut.

Both simulations use a mesh of 800 initial triangles.

5.2 Mesh Quality Comparison

We present a localized optimization to efﬁciently re-

mesh the ill-shape triangles and their one-ring neigh-

bors. Figure 8 compares the mesh after our cutting

simulation with the method that simply subdivides in-

tersected triangles with a static re-connect order (con-

ventional subdivision). Intuitively, the conventional

subdivision can easily generate ill-shaped triangles

Figure 6: Examples of cuts that (left) cut from a side and

(right) cut part of the mesh out. Note that each subdivided

mesh subsequently has an independent deformation basis

function.

Figure 7: High-detail cut surfaces which demonstrate in-

tersected cut (left) and multiple cuts within a single mesh

(right).

leading to unexpected cut edge distortion and abnor-

mal deformation. In contrast, our method can avoid

ill-shaped triangles, making the simulation numeri-

cally stable and visually convincing. Furthermore, we

locally increase the details only around the incision;

therefore the rest of the mesh maintains the level of

detail to ensure a focused computation cost.

Figure 9 illustrates how our adaptive subdivision

scheme generates optimal triangulation with low av-

erage aspect ratio (less than 3). We implemented the

conventional subdivision which simply subdividing

original triangle into three along the cut path on the

same size model. As shown in the ﬁgure, the con-

ventional subdivision can easily generate inefﬁcient

shaped or sized triangles with a high aspect ratio.

In contrast, our method optimizes all such triangles

whose aspect ratio is more than 3 to re-generate a sta-

ble and physically accurate incision.

5.3 Computation Cost Measurement

Table 1 shows the average computation times per sim-

ulation step for various levels of mesh resolution. We

apply an implicit time-based solver for the viscoelas-

tic deformation. The ﬁrst column represents the scale

of mesh, which followed by geometry modiﬁcation

cost that relates to the initial Delaunay triangulation

generation and localized optimization. The third col-

umn is the physics updating time after the re-meshing

GRAPP 2018 - International Conference on Computer Graphics Theory and Applications

188

Figure 8: With simply subdividing intersected elements

without further optimization (left), it leads to a swirling

cut edge. In contrast, our method (right) provides smooth,

physically accurate incisions.

Figure 9: Comparison of maximum aspect ratio after cut-

ting operation with conventional subdivision that without

any further optimization.

operation is completed. The last column shows the

total computation time of each cutting experiment.

The results indicate that our algorithm provides a

fast cutting process. Even in the highest quality model

(12800 triangles), the total computation time in each

call of our cutting method is less than six millisec-

onds. Moreover, the differential value of the total time

taken between the 800 triangles mesh and the 12800

triangles mesh is only 3.915 milliseconds, which sug-

gests that our cutting method can be integrated into a

high-quality model simulation in real-time.

Table 1: Performance measure (ms).

Triangles Geometry Solver Total Time

800 0.865 0.203 1.068

1800 0.947 0.278 1.225

3200 1.246 0.356 1.602

5000 1.987 0.390 2.377

7200 2.935 0.683 3.618

9800 3.598 0.702 4.30

12800 4.130 0.853 4.983

5.4 Level of Details Comparison

Figure 10 compares the level of details around the cut

incision with conventional subdivision (which only

subdivide the intersected triangles into three smaller

triangles along the cut path). Our method optimizes

the triangles generated by subdivision, with a small

number of additional DOFs.

Figure 10: Comparison of the level of detail around the inci-

sion with conventional subdivision scheme without further

subdivision.

6 CONCLUSION AND FUTURE

WORK

The cutting method described in this paper introduces

three novel contributions to the area of 3D simula-

tions of mesh cutting using FEMs. The ﬁrst signiﬁ-

cant contribution is the method of adopting the most

efﬁcient incremental Delaunay generation algorithm,

Bowyer-Watson algorithm, performing lower compu-

tational overhead than existing algorithms to generate

appropriate triangulation with low aspect ratio after

cutting occurs. Secondly, we achieved robust simula-

tion of complex cuts, such as re-cuts and intersected

cuts. Finally, we generate the optimal modiﬁed tri-

angulation by operating the local split optimization to

efﬁciently handle ill-shaped elements maintaining the

mesh quality with the maximum aspect ratio in the

triangulation of less than 3.

Our future work will focus on fusing the surface

model topology with the geometric model to produce

a component-based model to simulate detailed skin

model.

ACKNOWLEDGEMENTS

We thank all help and suggestion from Newcastle uni-

versity Game research group.

Accurate Real-time Complex Cutting in Finite Element Modeling

189

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