Summary
- We study irreversible evolutionary processes with a general
- energetic notion of stability. We dedicate this contribution to
- releasing three nonlinear variational solvers as modular components
- (based on FEniCSx/dolfinx) that address three mathematical
- optimisation problems. They are general enough to apply, in principle,
- to evolutionary systems with instabilities, jumps, and emergence of
- patterns which is commonplace in diverse arenas spanning from quantum
- to continuum mechanics, economy, social sciences, and ecology. Our
- motivation proceeds from fracture mechanics, with the ultimate goal of
- deploying a transparent numerical platform for scientific validation
- and prediction of large scale natural fracture phenomena. Our solvers
- are used to compute one solution to a problem encoded
- in a system of two inequalities: one (pointwise almost-everywhere)
+
We release three nonlinear variational solvers, developed as
+ modular components and built upon FEniCSx/dolfinx. These solvers
+ address mathematical optimisation problems for irreversible
+ evolutionary processes characterized by a general energetic notion of
+ stability, which we formulate via a well-posed variational principle.
+ These problems are general enough to apply, in principle, to systems
+ with instabilities, jumps, and emergence of patterns which is
+ commonplace in diverse arenas spanning from quantum to continuum
+ mechanics, economy, social sciences, and ecology. Our motivation
+ proceeds from fracture mechanics, with the goal of deploying a
+ transparent numerical platform for scientific validation and
+ prediction of large scale natural fracture phenomena. Our solvers are
+ used to compute one solution to a problem encoded in
+ a system of two inequalities: one (pointwise almost-everywhere)
constraint of irreversibility and one global energy statement. As part
of our commitment to open science, our solvers are released as free
software.
@@ -241,21 +242,8 @@ a Creative Commons Attribution 4.0 International License (CC BY
(Habera
& Zilian, n.d.) for parallel scalability.
Our solver’s API receives an abstract energy functional, a
- user-friendly description of the state of the system, its associated
- constraints, and the solver’s parameters. Solvers can be
- instantiated calling
- solver = {Hybrid,Bifurcation,Stability}Solver(
- E, # An energy (dolfinx.fem.form)
- state, # A dictionary of fields describing the system
- bcs, # A list of boundary conditions
- [bounds], # A list of bounds (upper and lower) for the state
- parameters) # A dictionary of numerical parameters
- where [bounds] are required for the
- HybridSolver, and used calling
- solver.solve(<args>) which triggers the
- solution of the corresponding variational problem. Here,
- <args> depend on the solver (see the
- documentation for details).
+ user-friendly description of the state of the system, and its
+ associated constraints.
HybridSolver solves a (first order)
constrained nonlinear variational inequality, implementing a
two-phase hybrid strategy which is ad hoc for
@@ -263,9 +251,7 @@ a Creative Commons Attribution 4.0 International License (CC BY
mechanics. The first phase (iterative alternate minimisation) is
based on a de-facto industry standard, conceived to
exploit the (partial, directional) convexity of the underlying
- mechanical models
- (Bourdin
- et al., 2000). Once an approximate-solution enters the
+ mechanical models. Once an approximate-solution enters the
attraction set around a critical point, the solver switches to
perform a fully nonlinear step solving a block-matrix problem via
Newton’s method. This guarantees a precise estimation of the
@@ -336,28 +322,32 @@ a Creative Commons Attribution 4.0 International License (CC BY
(error curve in blue). Note that the residual vector (green) for
the cone problem need not be zero at a
solution.
- ![]()
+ Rate of convergence for
+ StabilitySolver in 1d (cf. benchmark
+ problem below). Targets are the eigenvalue
+
+
+ limkλk=:λ*
+ (pink) and the associated eigen-vector
+
+
+ x*
+ (error curve in blue). Note that the residual vector (green) for
+ the cone problem need not be zero at a
+ solution.
Verification
- We benchmark our solvers against a nontrivial 1d problem
- (cf. test/test_rayleigh.py in the code
- repository), namely we compute
+ We benchmark our solvers against a nontrivial 1d problem, namely
+ we compute
minX0ℛ(z)andmin𝒦0+ℛ(z)[2],
using BifurcationSolver and
- StabilitySolver. The quantity
-
-
- ℛ(z)
- is a Rayleigh ratio, often used in structural mechanics as a
- dimensionless global quantity (an energetic ratio of elastic and
- fracture energies) which provides insight into the stability and
- critical loading conditions for a structure. For definiteness, using
- the Sobolev spaces which are the natural setting for second order
- PDE problems, we set
+ StabilitySolver. For definiteness, using the
+ Sobolev spaces which are the natural setting for second order PDE
+ problems, we set
X0=H01(0,1)×H1(0,1)
and
@@ -411,7 +401,23 @@ a Creative Commons Attribution 4.0 International License (CC BY
has support of size
D∈[0,1].
- ![]()
+ Comparison between profiles
+ of solutions
+
+ β(x)
+ in
+
+ X0
+ (left) vs.
+
+ 𝒦0+
+ (right). In the latter case, the solution
+
+
+ β(x)
+ has support of size
+
+ D∈[0,1].
The size of the support
@@ -425,7 +431,18 @@ a Creative Commons Attribution 4.0 International License (CC BY
vs. nontrivial solutions. Where error bars are not shown, the
error is
none.
- ![]()
+ The size of the support
+
+
+ D
+ for the minimiser in the cone. Error bars indicate the absolute
+ error. We observe a clear separation between constant solutions
+ with
+
+ D=1
+ vs. nontrivial solutions. Where error bars are not shown, the
+ error is
+ none.
Minimum value of
@@ -437,7 +454,16 @@ a Creative Commons Attribution 4.0 International License (CC BY
numerical computation vs. closed form result. Notice the
separation between constant solutions and nontrivial
solutions.
- ![]()
+ Minimum value of
+
+
+ ℛ
+ in
+
+ X0,
+ numerical computation vs. closed form result. Notice the
+ separation between constant solutions and nontrivial
+ solutions.
Minimum value of
@@ -457,24 +483,37 @@ a Creative Commons Attribution 4.0 International License (CC BY
R>1
are energetically
stable.
- ![]()
+ Minimum value of
+
+
+ ℛ
+ in
+
+ 𝒦0+,
+ numerical computation vs. closed form results. The outlier
+
+
+ (π2a,bc2)∼(4,0)
+ represents a computation which did not reach convergence. The
+ mechanical interpretation is that only states with
+
+ 1]]>
+ R>1
+ are energetically
+ stable.
-