MODELING IMPACTS WITH A CONSERVING AUGMENTED

MULTIBODY DYNAMICS 2007, ECCOMAS Thematic Conference
C.L. Bottasso, P. Masarati, L. Trainelli (eds.)
Milano, Italy, 25–28 June 2007
MODELING IMPACTS WITH A CONSERVING AUGMENTED
LAGRANGIAN METHOD
Juan C. Garcı́a Orden? , and Roberto A. Ortega?
? Departamento
de Mecánica de Medios Continuos y Teorı́a de Estucturas
Escuela de Ingenieros de Caminos, Universidad Politécnica de Madrid. c/ Profesor Aranguren s/n,
28040 Madrid, Spain
e-mails: [email protected], [email protected]
web page: http://w3.mecanica.upm.es
Keywords: Impact, Dynamics, Augmented Lagrangian, Conserving scheme
Abstract. This work analyses a conserving time integration scheme applied to impact dynamics of one or more points against a rigid surface. This rigid surface may represents the boundary
of a rigid body or it could represent a rigid cavity inside a body, and it should be represented
by an implicit function. The points may be material particles, vertex of a rigid polyhedra, or
nodes from a finite element discretization of continuous deformable body.
The non-penetration condition is enforced with an augmented Lagrangian technique, which
is applied after a regularization of the original unilateral constraint. The analysis of the presented methodology shows that exact enforcement of the non-penetration condition with exact
energy conservation is only achieved for some particular geometries and configurations prior
to contact. This analysis is supported by two numerical examples that illustrate these difficulties
and show the overall performance of the methodology.
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Juan C. Garcı́a Orden, Roberto A. Ortega
1
INTRODUCTION
Many engineering applications that use multibody models require an accurate and robust
representation of impacts. These impacts may occur during the normal operation of the system (e.g. mechanical actuators based on intermittent contacts among different parts) or may
appear in extraordinary situations, such as accidental external impacts, failure of some of the
components, etc. One interesting application of impact modeling in multibody systems is the
representation of joint clearances, due to geometrical imperfections of the connected parts, or
resulting from a progressive wear process [1, 13, 3, 4, 5].
In many of these aforementioned applications, a high number of multiple (in different places
of the system) and intermittent impacts appear among the different parts. Numerical long-term
simulations of such systems (in the time-scale of the rigid body motions of the different parts of
the mechanism) require an extraordinary robust handling of impacts [11, 12], allowing the use
of reasonable large time steps while getting also a reasonable accuracy on the representation of
each individual contact.
We propose in this work the use of a conserving time-stepping algorithm [14] with an augmented Lagrangian technique [2] to impose the non-penetration condition of the contact. This
contact is assumed to occur between two bodies, one of them at least being rigid and having
an analytical representation of the boundary surface. The boundary of the other body is represented by a discrete collection of facets, each one defined by points and edges (for instance, the
lagrangian mesh of a deformable body formulated with the finite element method).
Based on these considerations, the objective of the paper can be stated more precisely as the
development of a conserving augmented Lagrangian technique for the treatment of the dynamical contact (impact) of a point against an analytical surface which represents the boundary of
a rigid body. This approach imposes the exact non-penetration condition of the contact, while
conserving the total energy if no dissipative effects, such as viscous normal damping or tangential friction, are present. The main benefit of this energy conservation is the enhancement of the
stability of the discrete solution in time, allowing the use of large time steps.
The methodology described in this paper is based on previous results presented in [8] for
a conserving augmented Lagrange methodology applied to bilateral constraints, applied to the
modeling of permanent joints in mechanisms. The present paper extends these ideas to the
treatment of unilateral constraints, which is the base of the proposed robust formulation of
impact.
This technique can be effectively used in a wide range of applications involving a high number of impacts. For instance, it allows a robust representation of joint clearances with the socalled “cavity model”. In this model, one or more points of one of the bodies is constrained to
move inside a rigid cavity which is embedded in the other body. This type of representation is
useful to accurately represent clearances in spherical joints, or approximations to more complex
clearances. This approach is also amenable to be extended to clearances that are not properly
represented by the cavity model.
The article presents the detailed formulation of the proposed methodology, justifying their
conservation properties, and provides some insights about the specific numerical difficulties
posed by the augmented Lagrange procedure. Several representative numerical simulations are
presented, that serve to perform a discussion of the main performance features of the model.
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Juan C. Garcı́a Orden, Roberto A. Ortega
2
FORMULATION OF CONTACT
Let us consider the contact problem defined by point P and an implicit surface ϕ : R3 → R.
Point P may represent or be a part of a lot of different physical models. For instance, it may
represent a single material particle, it may also be a vertex of a polyhedron defining the boundary
of a rigid or deformable solid body (which is the case of a standard finite element model), etc.
On the other hand, surface ϕ encloses a rigid volume V. The position of point P relative to
the volume is obtained by a simple evaluation of the function ϕ at the point: ϕ = 0 if the point
is on the surface and ϕ < 0 or ϕ > 0 if the point is inside or outside the volume respectively.
Volume V which may also play different roles in the model; for instance:
• it may be bounded, representing a solid rigid body; for instance ϕ(x, y, z) = x2 /a2 +
y 2 /b2 + z 2 /c2 − 1 represents an ellipsoid in a cartesian coordinate system attached to it.
Denoting by r the position vector of point P relative to a reference system attached to the
body, the non-penetration condition of the rigid body is expressed as ϕ(r) ≥ 0, which
means that the point is constrained to move outside the volume, as shown in Figure 1.
ϕ
V
ϕ
V
P
P
ω
Figure 2: Rigid cavity: slider-crank mechanism with a clearance
Figure 1: Solid rigid body
• it may be bounded, representing a rigid cavity within a rigid or deformable body. In
this case, the non-penetration condition over point P is ϕ(r) ≤ 0 because the point
is constrained to move inside the volume. This is an useful model to represent some
type of joint clearances in multibody dynamics, where some freeplay exists between the
linked points of the mechanism. Figure 2 shows a slider-crank mechanism incorporating
a clearance of this type at the slider.
• it may be unbounded, representing a rigid frontier dividing the space. In this case, the
sign of ϕ determines the half-space where P lies. The point is constrained to move in one
side, and the non-penetration condition could be either ϕ ≤ 0 or ϕ ≥ 0. This is the case,
for instance, of a contact against a rigid wall represented by a plane.
All these cases share the same basic feature: contact is formulated with an inequality constraint involving the evaluation of the scalar function ϕ. The formulation of the dynamics of a
mechanism amenable to incorporate such contacts has some special characteristics:
1. It lead naturally to a set of differential-algebraic equations (DAE). The differential equations express the relation between the inertia forces, the applied forces and the constraint
forces introduced by the contact, while the algebraic equation express the non-penetration
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Juan C. Garcı́a Orden, Roberto A. Ortega
condition. The traditional approach is to formulate the contact force with a Lagrange multiplier λ, leading to a index-three DAE system. Its numerical solution often poses special
difficulties, mainly related to stability issues, very similar to the ones arising in stiff ordinary differential equation systems (ODEs).
2. The algebraic equation of the DAE system is an inequality, making it harder to solve compared with a similar case formulated with an equality relation, which would be the case
if the point were joined by a permanent link to the body. In practice, the difficulty of the
unilateral constraint is that contact has to be constantly checked, because the multiplier λ
only appears when the constraint is activated (it means, the point penetrates the surface).
3. Nevertheless, the contact check referred in the previous paragraph, which is a pure geometrical problem (determination of the geometrical interference between bodies) is simpler than the traditional point-to-surface that appears in models where all bodies are
bounded by polyhedral boundaries (e.g., traditional finite element models). In our case
contact detection relies in a simple evaluation of a function.
The methodology adopted in this work addresses the difficulties explained in items 1 and
2 of the previous list, while retaining the benefits of the simple contact detection as explained
in item 3. The procedure has two key ingredients: i) transform the inequality constraint into
an equality one, and ii) apply to the resulting formulation a conserving augmented Lagrangian
scheme to integrate in time.
Hopefully, the resulting scheme will exactly conserve the energy; but more important, it will
benefit of the good stability characteristics of a conserving integration while exactly enforcing
the non-penetration constraint. This objective can not always be successfully accomplished, as
it will be discussed in the next sections.
In order to explain these ideas, let us apply them to a simple case of the contact between a
particle P with mass m and a fixed solid rigid cavity V (similar to the case of 2) with boundary
defined by implicit function ϕ which it is at least C 1 . It will be assumed that there are not
applied forces other than the contact force between the particle and the cavity. Denoting by r
the vector of cartesian coordinates of P , the equations of motion have the general form:
mr̈ + fϕ = 0
ϕ(r) ≤ 0 ,
,
(1)
being fϕ the vector of contact forces, which is not null if the constraint is activated. Note that if
V represents a rigid solid instead of a cavity, the particle should be constrained to move outside
the volume; thus the constraint should change from ϕ ≤ 0 to ϕ ≥ 0. Nevertheless, it is clear
that the results obtained in the following sections for the cavity are general enough and can be
easily adapted for the case of a rigid solid.
As referred in the previous paragraphs, the first step of the methodology is to transform the
inequality into an equality constraint. We will call this process as constraint regularization.
Next, as an intermediate step, a conserving penalty scheme will be explained and applied to
the constrained problem. This explanation will ease the final description of the conserving
augmented Lagrangian scheme, which is the last ingredient of the method.
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Juan C. Garcı́a Orden, Roberto A. Ortega
2.1
Constraint regularization
The original constraint function ϕ is replaced by function Φ defined piecewise as:
(
0
if ϕ < 0
Φ = q 2 a+1
ϕ
if ϕ ≥ 0
a+1
(2)
with a regularization factor a ≥ 1. Note that:
• The problems formulated with the inequality ϕ ≤ 0 and Φ = 0 are equivalent, because
the volumes defined by ϕ ≤ 0 and Φ = 0 are the same.
• Function Φ has C 0 continuity. Besides, if ϕ is C 1 then Φ is, at least. piecewise C 1
continuous;
• Function Φ is null inside the volume (including the surface), and it is not null outside.
This means that Φ is globally C 1 only if its derivative at the surface is null.
Φ(x)
ϕ(x)
0
x
x0
a) Original ϕ(x)
0
Φ(x)
x
0
x0
x
x0
b) Φ(x) with a = 1
c) Φ(x) with a > 1
Figure 3: One-dimensional example with ϕ(x) = x − x0
• Based on the previous observation, and taking into account that the first derivative (the
gradient, denoted by (·)r = d(·)/dr) of Φ is expressed as:
(
0
if ϕ < 0
Φr = q a+1 a−1
(3)
ϕ 2 ϕr if ϕ ≥ 0
2
it becomes apparent that a > 1 guarantees the C 1 continuity of Φ for any function ϕ.
If a = 1, function Φ may have a discontinuity in the first derivative. This is illustrated
in Figure 3 with a one-dimensional example where the particle contacts with a rigid wall
located at x0 . In this case the original constraint is ϕ(x) = x − x0 , with a derivative which
is not null at the surface x = x0 , making Φ to be only C 0 for a = 1.
• Contact detection is still necessary, because the piecewise definition of function Φ, but it
still relies in a simple function evaluation.
3
CONSERVING TIME INTEGRATION SCHEMES
The equations of motion of the contact problem (1) are now stated in terms of the regularized
constraint as:
mr̈ + fΦ = 0 ,
Φ(r) = 0 ,
(4)
where fΦ is the contact force associated to the new constraint Φ. Problem (4) may be solved with
several numerical techniques; one of such techniques is a conserving penalty scheme, which is
briefly presented next.
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Juan C. Garcı́a Orden, Roberto A. Ortega
3.1
Conserving penalty scheme
Based on the early works of Simó & Tarnow [14] and González & Simó [10] on conservative integrators for nonlinear elastodynamics, it is possible to formulate a second-order discrete
scheme for (4) that exactly enforces the conservation of the total mechanical energy and the
linear and angular momenta. The main advantage of this approach, apart from the desirable
discrete conservation of primary physical magnitudes of the motion, is the remarkable improvement of the numerical stability for non-linear problems.
For a constraint Φ(r) which it is at least C 1 , the conserving penalty discrete time-marching
scheme for problem (4) takes the form (see details in [6, 9, 7]) :
m (ṙn+1 − ṙn ) + ∆t α ΦT
rn+β Φn+ 12 = 0
1
1
(ṙn+1 + ṙn ) =
(rn+1 − rn )
2
∆t
(5)
where:
• ∆t = tn+1 − tn is the time step;
• α is the penalty parameter, which influence the precision of the constraint fulfillment,
making Φ → 0 when α → ∞;
• subscript (·)n+β means evaluation of magnitude (·) at the intermediate configuration
def
rn+β = rn + β(rn+1 − rn ), with β ∈ [0, 1]
• parameter β should satisfy the following (typically non-linear) scalar equation:
Φrn+β (rn+1 − rn ) = Φn+1 − Φn
(6)
The proof of existence of a solution β ∈ [0, 1] of (6) is based on the mean-value theorem
applied to function Φ. This is the reason because continuity C 1 is required.
• the overline of the term Φn+ 1 denotes the average of a magnitude between tn and tn+1 ,
2
def
such that (·)n+ 1 = [(·)n + (·)n+1 ]/2
2
It can be shown [6, 9, 7] that scheme (5) exactly enforces conservation of total energy between tn and tn+1 , which is the sum of the kinetic energy T = (1/2)mṙ 2 and the constraint
energy VΦ = (1/2)αΦ2 . Physically, the constraint stores a portion of the energy when penetration occurs (penetration is always not null if penalty is employed) and releases this energy
exactly when contact finishes and the particle is again inside the cavity.
One drawback of this methodology is that the non-penetration condition is not exactly satisfy. The other main drawback is that small penetrations are only achieved using large penalty
parameters, which causes the equation system to be stiff. Nevertheless, this last issue is satisfactory handled by the conserving integration, which has an extraordinary numerical stability.
The failure of the non-penetration condition motivates the development presented in the next
section, which combines an augmented Lagrangian technique with a conserving time integration.
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Juan C. Garcı́a Orden, Roberto A. Ortega
3.2
Conserving augmented Lagrangian scheme
In this technique we add a new term to the penalty formulation (5) containing a Lagrange
multiplier λ. The discrete time-marching scheme takes then the form:
T
m (ṙn+1 − ṙn ) + ∆t Φrn+β αΦn+ 1 + λ = 0
(7)
2
1
1
(ṙn+1 + ṙn ) =
(rn+1 − rn ) ,
2
∆t
where parameter β ∈ [0, 1] should satisfy equation (6), and it always exists provided that the
constraint Φ has at least C 1 continuity.
The usual implementation of this scheme is based on the Uzawa’s method, which sets an
iterative procedure for the calculation of the Lagrange multiplier λ. For non-linear problems,
its usual to use a nested iteration, where λ is kept constant in (7), which is solved for rn+1 and
ṙn+1 . Afterwards, the Lagrange multiplier is updated with the formula:
λ(k+1) = λ(k) + αΦn+ 1 ,
2
(8)
where the superscript (·)(k) denotes the value of (·) at the iteration k of the loop on λ.
Note that the whole scheme given by (7) and (8) has two nested loops: the outer one is for λ,
and the inner loop is typically a Newton-Raphson type iteration for the solution of the non-linear
equations (7), where λ is kept constant.
It is possible to show that the scheme defined by (7), (8) and (6) exactly conserves the total
mechanical energy while exactly enforcing constraint Φ (see detailed proofs in [8]), provided
that Φ has at least C 1 continuity.
Note that the conserved energy does not include terms due to the constraint, as it was the
case of the penalty formulation. In this case the total energy is the sum of the kinetic energy and
the potential energy of the applied conservative forces, which are null for the specific system
defined by equations (1).
If constraint Φ depends on the modulus of the position vector ||r|| = r instead of a general
expression on the coordinates of r, the conservative formulation does not involve a calculation
of the parameter β, and so it does not require Φ to have C 1 continuity. We will denote this as
an scalar constraint, and it is, for example, the case of an spherical cavity of radius r0 , defined
by the implicit surface ϕ(r) = r − r0 . The conservative scheme is then given by:
Φn+1 − Φn
1 + λ
1
r
αΦ
=0
n+ 2
2
rn+1
− rn2 n+ 2
1
1
(ṙn+1 + ṙn ) =
(rn+1 − rn )
2
∆t
m (ṙn+1 − ṙn ) + 2∆t
(9)
with the same update (8) for the Lagrange multiplier (see details in [8]).
This formulation (either (7) or (9)) has been successfully applied in [8] to mechanical systems subjected to general bilateral constraints Φ = 0. In these applications, constraints are not
derived from a regularized unilateral constraint ϕ ≤ 0 as in (2)), but represent permanent links
among different points of the system; e.g., a constant distance constraint representing a rigid
link between two points or a prescribed path that a point has to follow in a three-dimensional
space.
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Juan C. Garcı́a Orden, Roberto A. Ortega
4
CONSERVING AUGMENTED LAGRANGIAN CONTACT
Based on the considerations of the previous section, it becomes apparent that the scheme
given by (7), (8) and (6) may provide a suitable model for contact, provided that constraint Φ
defined by (2) represents the non-penetration condition for a given contact surface ϕ.
However, some practical difficulties arise in such formulation due to some special characteristics of the regularized constraint function Φ. Some of them have been mentioned in section
2.1, and may even prevent from being able to define a conserving augmented Lagrange formulation, as it will be shown in the next paragraphs.
4.1
General surface
Let us consider the case of a general surface ϕ(r), and its associated regularized constraint
Φ(r) defined by (2). We will assume that the particle is either inside the cavity or exactly on
the surface at tn (it means, ϕ(rn ) ≤ 0) and it has a velocity ṙn such that it would penetrate the
surface at tn+1 if no contact is enforced, as shown in Figure 4.
ϕ=0
(0)
tn+1
tn+1
(k)
tn+1
ϕ<0
tn
tn
Figure 4: Configurations at tn and tn+1 without
contact enforcement
Figure 5: Sequence as iteration in λ progresses
According to the results presented in section 3.2, it is possible to use the conserving augmented Lagrangian scheme given by (7) and (8) provided that the regularized function Φ is at
least C 1 . This implies that the gradient Φr evaluated at the surface ϕ = 0 should be null. This
is the situation for a regularization factor a strictly greater than 1, as illustrated in the example
of Figure 3 of section 2.1.
Now, let us now take a closer look at the conserving formulation (7). The constraint force f Φ ,
may be written as the sum of two forces related to the penalization and the Lagrange multiplier
respectively:
m(ṙn+1 − ṙn ) + ∆t fΦ = 0
,
T
with fΦ = ΦT
rn+β αΦn+ 12 + Φrn+β λ
|
{z
} | {z }
fλ
fα
(10)
Note that force fΦ should converge to a finite and not-null magnitude as k → ∞, because it is
proportional to the balance of linear momentum of the particle during the impact.
Let us discuss the behavior of the different terms in (10) as the iteration in λ progresses, with
(0)
k → ∞. Note that the particle moves from the predicted configuration tn+1 towards the surface,
8
Juan C. Garcı́a Orden, Roberto A. Ortega
(k)
as shown schematically in Figure 5. This means that Φn+1 → 0 as k → ∞. Taking into account
that the particle was inside the cavity at tn and therefore Φn = 0, this also means that:
(k)
Φn+ 1 → 0
as
k → ∞ (iteration in λ progresses)
(11)
2
On the other hand, the Jacobian Φr evaluated at n + β verifies equation (6), provided that Φ
is at least C 1 thus guaranteeing the existence of β. Taking into account again that Φn = 0 and
(k)
Φn+1 → 0, from (6) it follows that:
Φ(k)
rn+β → 0
as
k→∞
(12)
Taking into account relations (10), (11) and (12), it becomes apparent that fα → 0 and λ(k)
should tend to ∞ as (k) → ∞ for a bounded and not-null contact force fΦ → fλ . This means
that it is impossible for the loop on λ to achieve convergence. Thus, scheme (7) does not work
properly, failing to exactly enforce the non-penetration condition while exactly conserving the
energy.
Note that this difficulty would disappear if it would be possible to remove the condition of
Φ of being C 1 . In this case we could perform the regularization of ϕ given by (2) with a = 1.
Introducing a = 1 in (3), the Jacobian results:
(
0
if ϕ < 0
Φr =
(13)
ϕr
if ϕ ≥ 0
Thus, Φrn+β = ϕrn+β could tend to a finite value different from null as the point get closer
to the surface. This happens, for instance, if the original surface ϕ is convex, and it is also
the case of the example of Figure 3. As a consequence, the loop on the Lagrange multiplier λ
would converge to a finite value, the contact force would be then fΦ = fλ and the scheme would
apparently perform well.
Nevertheless, the previous arguments are useless since the C 1 continuity of Φ is a necessary
ingredient of the formulation, allowing the calculation of parameter β responsible of the exact
energy conservation. Summarizing, energy conservation demands a regularization with a > 1
which is incompatible with the convergence of the Lagrange multiplier, thus failing to impose
an exact non-penetration on the surface.
4.2
Spherical surface
This case differs from the previous one, because the conserving augmented Lagrange formulation is possible in certain situations, as it will be shown in the next paragraphs.
A spherical surface is expressed, in general, by a function ϕ which depends only on the
modulus of the position vector ||r|| = r. As described in section 3.2, its conserving augmented
Lagrange formulation does not require the C 1 continuity of Φ, and it is given by equations (9).
Similar to the procedure followed in section 4.1, the constraint force f Φ may be written in terms
of two forces related to the penalization and the Lagrange multiplier respectively:
m(ṙn+1 − ṙn ) + ∆t fΦ = 0 ,
Φn+1 − Φn
Φn+1 − Φn
with fΦ = 2 2
αΦn+ 1 rn+ 1 + 2 2
λrn+ 1
2
2
2
2
rn+1 − rn
rn+1 − rn2
|
{z
} |
{z
}
fα
fλ
(14)
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Juan C. Garcı́a Orden, Roberto A. Ortega
Following the same arguments provided in section 4.1 with Figure 5, the configuration at
get closer to the surface as the iteration in λ progresses, which means that:
(k)
tn+1
(k)
(k)
Φn+1 → 0 ,
Φn+1 − Φn → 0 ,
(k)
Φn+ 1 → 0
as
k→∞
2
(k)
If the particle is out of the surface at tn , as shown in Figure 5, it follows that (rn+1 −rn ) 6→ 0,
because the iteration in λ moves the particle towards the surface, but it does not surpasses it.
On the other hand, fΦ should tend to a not-null and bounded value, since it is proportional
to the linear momentum of the particle. Taking into account these considerations, it becomes
apparent that fα → 0 and the value of λ should tend to ∞, precluding its iteration loop to
converge. Thus, if the particle is not on the surface at tn , the scheme (9) fails to exactly satisfy
the non-penetration condition while conserving the total energy.
(k)
A different situation arises if the particle is on the surface at tn . In this case (rn+1 − rn ) → 0
as iteration in λ progresses, which causes an indetermination in the terms of f α and f λ that
should be solved. Taking into account that Φn = 0, the penalty term is given by:
fα = α
Φ2n+1
0
rn+ 1 −→ rn
2
2
2
rn+1 − rn
0
rn+1 → rn
as
Employing the LHôpital’s rule for solving this indeterminate relation gives:
lim
+
rn+1 →rn
2Φn+1 Φ0n+1
Φ2n+1
= lim +
=0
2
rn+1
− rn2
2rn+1
rn+1 →rn
(15)
def
where (·)0 = d(·)/dr is the derivative from the right. Note that the limit has been taken from
the right, because rn+1 converges to rn towards the cavity, as shown in Figure 5. This implies
that the derivative Φ0n+1 in (15) is also taken from the right. Thus, function Φ could not have a
derivative defined right at the surface, but it suffices to have a bounded derivative from the right,
as rn+1 → rn+ . From this considerations, it can be concluded that the penalty term fα → 0 as
the particle goes back to the surface.
The force with the Lagrange multiplier for Φn = 0 is given by:
fλ = 2λ
Φn+1
r 1
2
rn+1 − rn2 n+ 2
0
−→ 2λ∗ rn
0
as
rn+1 → rn
(16)
being the λ∗ the converged Lagrange multiplier. Using again LHôpital’s rule for solving this
indeterminate relation gives:
lim
+
rn+1 →rn
Φ0n+1
Φn+1
=
lim
,
2
+ 2r
rn+1
− rn2
rn+1 →rn
n+1
(17)
which is not-null and bounded provided that the derivative from the right of Φ at the surface
exists and it is not null. This the case when a regularization parameter a = 1 is employed and
the derivative from the right of the original function ϕ is not null at the surface, as illustrated by
the example of Figure 3 b). Note that a = 1 may cause Φ to loose its C 1 continuity, but this is
not a difficulty in this case since the conserving scheme (9) does not require it.
Summarizing, only if: i) a regularization factor a = 1 is employed; ii) ϕ0 6= 0 (derivative
from the right) at the surface; and iii) the particle is at tn is on the surface, the scheme (9)
exactly enforces the non-penetration condition while conserving the total energy.
The concepts presented in the previous sections are illustrated in the two numerical examples
presented in the next section.
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Juan C. Garcı́a Orden, Roberto A. Ortega
5
NUMERICAL EXAMPLES
Two numerical examples are presented next. The first one consists in a particle moving
inside a spherical cavity represented by a scalar constraint, and the second is similar to the
previous one, but in this case the cavity is defined by an elliptical surface represented by a
general constraint. In both cases the movement of the particle is obtained by means of the
conserving augmented Lagrange schemes presented in previous sections.
5.1
Spherical cavity
The system is composed by one particle with mass m = 1 kg with a position parametrized
by its two cartesian coordinates r = (x, y)T . The particle is constrained to move inside a fixed
spherical cavity defined by function ϕ, which has a regularized constraint function Φ given by:
r
2
Φ(r) =
ϕa+1 , with
ϕ(r) = r − r0 (spherical surface)
(18)
a+1
p
being r = ||r|| = x2 + y 2 the distance from the origin of the cartesian coordinate system,
and r0 = 0.5 m. Thus, contact detection is performed simply evaluating function ϕ(r) for a
given distance r of the particle from the center of the cavity. The initial position and velocity of
the particle are r0 = (0.25, 0.10)T m and ṙ0 = (0.5 cos α, 0.5 sin α)T m/s, as shown in Figure
6. Gravity is not considered.
State
0.5
State I
0.4
tn+1
tn
y
(m)
4
α
0.3
3
State II
0.2
1
α
1
0.1
0
I
0
0.1
tn−1
r0
State
ṙ 0
α
r0
0.2
x
0.3
(m)
II
tn+1
0.4
tn
0.5
tn−1
Figure 6: Geometry of contact
Figure 7: Graphic description of states I and II
A variation of the angle α, which defines the orientation of the initial velocity, allows to
generate two different situations of contact. One is denominated state I and the other is called
state II. In state I, the particle is located on the surface when the contact begins, while in state
II this does not happen. Figure 7 illustrate this two situations.
11
Juan C. Garcı́a Orden, Roberto A. Ortega
The movement is numerically integrated in time with the conserving scheme (9), suitable for
a scalar constraint such as (18) with a time step ∆t = 0.025 s. The penalty parameter is set to
α = 104 .
Figures 8 and 9 shows the evolution of the constraints forces when the iteration in the Lagrange multiplier progresses. In order to analyze the state I two possible values for the regularization parameter a = 1 and a = 1.5 are considered.
Time step:
0.025
Time step:
s.
40
a=1
35
||f λ ||
||f λ ||
30
Fore (N)
Fore (N)
s.
a>1
35
30
25
20
15
10
25
20
15
10
||fα ||
5
0
0.025
40
0
2
4
6
8
Iterations
10
5
0
12
Figure 8: State I. Constraints forces vs. λ iterations
||fα ||
0
5 10 15 20 25 30 35 40 45 50
Iterations
Figure 9: State I. Constraints forces vs. λ iterations
Figure 10 shows the evolution of the Lagrange multiplier when the iteration in the multiplier
progresses, and for the two different a parameters. Figure 11 shows the evolution of the total
energy E, which equals in this case the kinetic energy T = (1/2)mṙ 2 .
Time step:
0.025
s.
Time step:
110
a>1
100
Energy (J)
80
λ
70
60
50
40
a=1
0.1
0.08
0.06
0.04
a=1
a>1
0.02
20
0
10
0
2
4
6
8
10
s.
E=T
0.12
90
30
0.025
0.14
12
Iterations
Figure 10: State I. Lagrange multiplier vs. λ
iterations
0
2
4
6
8
Iterations
10
12
Figure 11: State I. Energy vs. λ iterations
It has to be remarked also that the convergence requirement adopted for the Lagrange multipliers in this example is applied on the penalty constraint energy VΦ = 12 αΦ2 ≤ 10−10 .
12
Juan C. Garcı́a Orden, Roberto A. Ortega
Figure 12 shows that the Lagrange formulation effectively enforces the constraint with the
parameter a = 1, as studied in the previous sections. The results with parameter a > 1 show
that the scheme fails to converge, as it was predicted by the theoretical results. Figure 13 shows
the trajectory of the particle up to tf = 1 s, where the contact event can be observed.
Time step:
0.025
Time step:
s.
0.025
s.
0.5
5
Φ × 10−3
Φ × 10−3
4.5
4
a=1
0.4
3.5
(m)
y
Φ
3
2.5
a>1
2
0.3
tf = 1
0.2
1.5
0.1
1
t0 = 0
a=1
0.5
0
0
0
2
4
6
8
10
12
0
0.1
0.2
x
Iterations
Figure 12: State I. Constraint vs. λ iterations
0.3
(m)
0.4
0.5
Figure 13: State I. Trajectory vs. Time
Now we analyze the results for the state II, considering just one case with a regularization
parameter a = 1. This is a interesting case because it was the one that performed well for
state I, when the particle was right at the surface in tn . Figure 14 shows the evolution of the
constraint forces when the iteration in the Lagrange multiplier progresses during the contact. It
shows that the augmented Lagrange formulation is not possible in this case, as already justified
in the previous sections. The convergence in the Lagrange multiplier is not obtained, with a
behavior that is very similar to the previous case (state I with parameter a > 1).
a=1
Time step:
0.025
s.
30
a=1
Time step:
0.025
s.
0.022
0.018
20
15
10
0.016
0.014
0.012
0.01
0.008
5
0
E=T
0.02
||f λ ||
Energy (J)
Fore (N)
25
||fα ||
0
0.006
0.004
5 10 15 20 25 30 35 40 45 50
Iterations
Figure 14: State II. Constraints forces vs. λ
iterations
0
5 10 15 20 25 30 35 40 45 50
Iterations
Figure 15: State II. Energy vs. λ iterations
13
Juan C. Garcı́a Orden, Roberto A. Ortega
a=1
Time step:
0.025
s.
a=1
250
Time step:
0.025
s.
5
Φ × 10−3
4.5
200
4
3.5
150
Φ
λ
3
100
2.5
2
1.5
50
1
0.5
0
0
5
0
10 15 20 25 30 35 40 45 50
0
Iterations
10 15 20 25 30 35 40 45 50
Iterations
Figure 16: State II. Lagrange multiplier vs. λ
iterations
5.2
5
Figure 17: State II. Constraint vs. λ iterations
Elliptical cavity
In this example the conserving augmented Lagrange formulation is analyzed for a general
constraint:
r
2
x2 y 2 z 2
Φ(r) =
(19)
ϕa+1 , ϕ(r) = 2 + 2 + 2 − 1
a+1
cx
cy
cz
which represents an elliptical cavity centered at the
origin of coordinates, with cx = cy = 0.3 m and
cz = 0.6 m. The regularization of this constraint is
made with parameter a = 4. The system composed
by one particle with mass m = 1 kg with a position parametrized with tree cartesian coordinates
r = (x, y, z)T . The contact detection is performed
ṙ 0
in this case by a simple evaluation of function ϕ(q).
The initial position and velocity are given by r0 =
r0
(0, 0, 0)T m and ṙ0 = (0.4, 0.4, 0.4)T m/s, as shown
cz
in Figure 18. Gravity is not considered.
The movement is integrated with a conservative
scheme with a time step ∆t = 0.025 s. The penalty
cx
parameter is set to α = 107 . The convergence recy
quirement adopted for the Lagrange multipliers in
this example is applied on the penalty constraint energy VΦ = 21 αΦ2 ≤ 10−10 .
Figures 19 and 20 shows the constraint forces
Figure 18: Elliptical cavity
and the energy respectively. It can be observed that
the convergence of the Lagrange multipliers is not
achieved; while the force fα is goes to zero, the force fλ always increases, although the energy
does not completely recover.
Figures 21 and 22 shows the evolution of the Lagrange multiplier and the constraint respectively. The multiplier always increases and the constraint goes slowly to zero when the Lagrange
multiplier iterations progresses, illustrating the non-convergence of the method.
14
Juan C. Garcı́a Orden, Roberto A. Ortega
a>1
Time step:
0.025
s
a>1
45
35
0.2
30
0.18
25
20
15
0.16
0.14
0.12
0.08
5
0.06
||fα ||
0
0.04
5 10 15 20 25 30 35 40 45 50
Iterations
8000
a > 1 Time step: 0.025
7000
6000
5000
Φ
4000
3000
2000
1000
0
0
10
20
30
Iterations
40
50
Figure 20: Energy vs. λ Iterations
Figure 19: Constraints forces vs. λ iterations
λ
s
0.1
10
0 5 10 15 20 25 30 35 40 45 50
Iterations
Figure 21: Lagrange multiplier vs. λ iterations
6
0.025
E=T
0.22
||f λ ||
Energy (J)
Fore (N)
40
0
Time step:
0.24
20
18
16
14
12
10
8
6
4
2
0
a > 1 Time step: 0.025
Φ × 10−5
0 5 10 15 20 25 30 35 40 45 50
Iterations
Figure 22: Constraint vs. λ iterations
CONCLUSIONS
A methodology that combines an augmented Lagrangian formulation of contact and a conserving time integration scheme has been presented. An analysis of this methodology applied
to the impact of a particle against a rigid surface has been carried out, with the following main
results:
• The methodology has been probed to perform very well for bilateral constraints, representing permanent links on multibody systems [8].
• In order to use this methodology to model contact, it is necessary to extend these ideas
to the representation of unilateral constraints. This is accomplished performing a constraint regularization, which transforms the original inequality constraint to an equality
constraint.
• The regularized function has some peculiarities that affect both the applicability and the
15
Juan C. Garcı́a Orden, Roberto A. Ortega
performance of the scheme. As a result, exact non-penetration and exact energy conservation are only achieved for spherical surfaces, and only when the point lies right on the
surface when the contact occurs.
• This result is a serious limit to the presented methodology. Based on this, we address at
this moment the recomendation of employing a conserving penalty formulation in case
of non-spherical geometries, which leads to a very robust modelling of multiple impacts,
as reported in [5]. For spherical surfaces it may be attactive to employ the conserving
augmented Lagrange formulation provided that some time step adaptativity is adopted,
such that the point right before the contact is located as close as possible to the surface.
ACKNOWLEDGEMENTS
Both authors want to acknowledge the support by the Ministerio de Educación y Ciencia
from the Government of Spain with the project DPI2006-15613-C03-02 entitled “Modelización
numérica eficiente de grandes sistemas flexibles con aplicaciones de impacto”.
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