Efficient solvers for time-dependent problems: a review of IMEX, LATIN, PARAEXP and PARAREAL algorithms for heat-type problems with potential use of approximate exponential integrators and reduced-order models <s> Iterative methods: the LATIN approach <s> 1 The Reference Problem for Small Disturbances.- 1.1. Notation.- 1.2. The reference problem.- 1.3. Sufficient conditions assuring uniqueness.- 1.4. Analogy with the basic problem of fluid mechanics.- 2 Material Models.- 2.1. Formulation with internal variables.- 2.2. Examples of material models.- 2.3. Formulation of the constitutive relation.- 2.4. Normal formulation of a constitutive model.- 2.5. Error as measured by the constitutive relation (error in CR).- 2.5.1. Some classical properties of constitutive models.- 2.5.2. Error measures for a standard normal formulation.- 2.5.3. Illustration of the notion of admissibility as regards internal variables.- 2.5.4. Error in the sense of Drucker-functional formulation.- 2.5.5. Extensions.- 3 Solution Methods for Nonlinear Evolution Problems.- 3.1. The principle of incremental methods.- 3.2. Differential equation formulation of the reference problem.- 3.3. A general presentation of some classical methods for solving nonlinear problems.- 3.3.1. The geometric scheme associated with the problem.- 3.3.2. Algorithms for two search directions.- 3.3.3. Description of the different stages.- 3.3.4. Examples of directions of descent and ascent.- 3.3.5. Errors and error indicators.- 3.3.6. A convergence result.- 3.4. Other approaches to nonlinear evolution problems.- 4 Principles of the Method of Large Time Increments.- 4.1. Mechanics framework for the method of large time increments.- 4.2. Algorithms for two search directions.- 4.3. The local step.- 4.3.1. General case.- 4.3.2. Examples: plastic and viscoplastic materials with isotropic hardening.- 4.4. The global linear step.- 4.4.1. Quasi-static linear global step.- 4.4.2. The linear global step in dynamics.- 4.5. Convergence.- 4.5.1. Principal hypotheses.- 4.5.2. Basic identities.- 4.5.3. Convergence results.- 4.6. A posteriori error estimates.- 4.6.1. A first set of error indicators.- 4.6.2. Analysis of the caseH+=H-=L(Lsymmetric and positive).- 4.6.3. Other error indicators.- 4.7. Remarks.- 5 A Preliminary Example: A Beam in Traction.- 5.1. Quasi-static analysis for a viscoplastic material.- 5.2. Static analysis for a hyperelastic material.- 6 A "Mechanics Approximation" and Numerical Implementation.- 6.1. Discretization in time and space.- 6.2. Numerical treatment of the local step.- 6.3. Treatment of the linear global step in statics.- 6.3.1. Approximation on ? x [0T] (Principle P3).- 6.3.2. Iterative method for solving the linear global step.- 6.3.3. Remarks.- 6.4. Decomposition and approximation of the "radial loading" type for a function defined on ? x [0T].- 6.4.1. Approximation of order 1.- 6.4.2. Properties of the associated eigenvalue problem.- 6.4.3. Approximation of ordermand convergence properties.- 6.4.4. Remarks.- 6.5. Applications and analysis of performance.- 6.5.1. Example 1.- 6.5.2. Example 2.- 6.5.3. Example 3.- 7 Modeling and Calculation for Structures under Cyclic Loads.- 7.3. Treatment of the linear global step.- 7.4. A one-dimensional example.- 7.5. Example: viscoplastic disk with a loading of 1,000 cycles.- 8 Formulation and "Parallel" Strategies in Mechanics.- 8.1. Remarks on the degree of parallelism in the equations of reference.- 8.2. Partioning of the body into sub-structures and interfaces.- 8.2.1. Principles of the partioning method.- 8.2.2. Examples of interfaces.- 8.2.3. Modeling of an interface.- 8.2.4. New formulation with partitioning of the reference problem.- 8.3. Treatment of a static assemblage of elastic structures.- 8.3.1. Formulation of the problem.- 8.3.2. The local step: $${{s}_{n}} \to {{\hat{s}}_{{n + \frac{\hbox{$\scriptstyle 1$}}{\hbox{$\scriptstyle 2$}} }}}$$.- 8.3.3. The semi-global linear step: $${{\hat{s}}_{{n + \frac{\hbox{$\scriptstyle 1$}}{\hbox{$\scriptstyle 2$}} }}} \to {{s}_{{n + 1}}}$$.- 8.3.4. Example.- 8.4. Convergence for a static assemblage of elastic structures.- 8.4.1. Principal hypotheses.- 8.4.2. A preliminary convergence result ( a positive constant).- 8.4.3. Convergence results for = 0.- 8.5. Dynamic and static treatment of an assemblage of structures with nonlinear behavior.- 8.5.1. A new formulation with partitioning of the reference problem.- 8.5.2. The local step $${{s}_{n}} \to {{\hat{s}}_{{n + \frac{\hbox{$\scriptstyle 1$}}{\hbox{$\scriptstyle 2$}} }}}$$.- 8.5.3. The linear step (semi-global) $${{\hat{s}}_{{n + \frac{\hbox{$\scriptstyle 1$}}{\hbox{$\scriptstyle 2$}} }}} \to {{s}_{{n + 1}}}$$.- 8.5.4. Convergence of the method.- 9 Modeling and Computation for Large Deformations.- 9.1. Material quantities and modeling of their behavior.- 9.2. Pure material formulation of large deformations-bases.- 9.3. Kinematic and other properties.- 9.3.1. Calculation of A as a function of $$\dot{\Sigma }$$.- 9.3.2. Calculation of Q as a function of $$\dot{\Sigma }$$.- 9.3.3. Calculation of ? and R as functions ofV.- 9.3.4. Other properties.- 9.4. Purely material formulation of the equilibrium of the body-properties and approximations.- 9.4.1. Reference formulation.- 9.4.2. The approximation A~1.- 9.4.3. The notion of radial loading.- 9.4.4. The problem in velocity.- 9.5. Two different representations of the modeling and computation of large deformations.- 9.5.1. Presentation with "linear" equilibrium equations.- 9.5.2. Presentation with "nonlinear" equilibrium equations.- 9.6. Approaches to large time increments.- 9.6.1. A first approach.- 9.6.2. Large time increment approaches to constitutive models with internal variables.- 9.6.2.1. Presentation with "linear" equilibrium equations.- 9.6.2.2. Presentation with "nonlinear" equilibrium equations.- 9.7. Remarks and an example. <s> BIB001 </s> Efficient solvers for time-dependent problems: a review of IMEX, LATIN, PARAEXP and PARAREAL algorithms for heat-type problems with potential use of approximate exponential integrators and reduced-order models <s> Iterative methods: the LATIN approach <s> This paper deals with the synergy between the LATIN multiscale method and what is called the Proper Generalized Decomposition (PGD) which is the key of its performances. <s> BIB002
A usual way to deal with nonlinear equations numerically is to use an iterative process within a fixed point algorithm. The LATIN (LArge Time INcremental) method pioneered by Ladevèze BIB001 and since then broadly used in computational structural Mechanics and material science (see BIB002 for a recent reference) solves time-dependent problems (linear or nonlinear) according to a two-step iterative process. To separate the difficulties, equations are partitioned into two groups: (i) a group of equations being local in space and time, possibly nonlinear (representing equilibrium equations for example); (ii) a group of linear equations, possibly global in the spatial variable. Then ad-hoc space-time approximations methods are used for the treatment of the global problem. Of course, space-time local equations can be solved in parallel, what makes the LATIN method efficient and suitable for today's HPC facilities. Let us emphasize that with LATIN, it is possible to solve hard nonlinear mechanics problems including thermodynamics irreversible problems (plasticity, friction as examples). As an llustration, let us describe the LATIN method on the (rather simple) nonlinear heat problem: and time) of the nonlinear problem (it can be an approximate solution obtained with a coarse solver for example); computeκ (0) = κ(u (0) ); 2. Iterate k, step 1 (global linear solution). Solve the linear problem with given initial and boundary conditions. 3. Iterate k, step 2 (local projection over the admissible manifold). Computẽ 4. Check convergence, k ← k + 1 if not and go to 2. The step 1 performs a global (linear) evolution of the solution whereas a pointwise nonlinear projection on the equilibrium conductivity coefficients is done in step 2. We have a natural convergence indicator in terms of distance between the frozen conductivitỹ κ and κ(u (k) ): In particular, if κ is Lipschitz continuous with Lipschitz constant L, then Remark that step 2 can be performed in parallel (in time). For better and faster convergence, one can imagine variant approaches using a relaxation approach: remark first that κ(u) is (formally) solution of the partial differential equation If κ is a strictly convex function for example, then the second term at the right hand side is negative. One may consider the approximate (augmented) problem where ε > 0 is a given relaxation time (assumed to be rather small). By this way, it is expected thatκ evolves much closer toward the value κ(u). One can then derive an iterative process with again two steps (linear solution + projection) as in the LATIN method.