Discontinuity layout optimization
Encyclopedia
Discontinuity layout optimization (DLO) is an engineering analysis
procedure which can be used to directly establish the amount of load
that can be carried by a solid or structure prior to collapse. Using DLO the layout of failure planes, or 'discontinuities', in a collapsing solid or structure are identified using mathematical optimization methods (hence the name, 'discontinuity layout optimization'). It is assumed that failure occurs in a ductile or 'plastic
' manner.
The set of potential discontinuities can include discontinuities which crossover one another, allowing complex failure patterns to be identified (e.g. involving ‘fan’ mechanisms, where many discontinuities radiate from a point).
DLO can be formulated in terms of equilibrium relations ('static' formulation) or in terms of displacements ('kinematic' formulation). In the latter case the objective of the mathematical optimization problem is to minimize the internal energy dissipated along discontinuities, subject to nodal compatibility constraints. This can be solved using efficient linear programming
techniques and, when combined with an algorithm originally developed for truss layout optimization problems, it has been found that modern computer power can be used to directly search through very large numbers of different failure mechanism topologies (up to approx. 21,000,000,000 different topologies on current generation PCs). A full description of the application of DLO to plane strain problems has been provided by Smith and Gilbert
procedure, mathematical relations are formed for the underlying continuum mechanics problem, DLO involves analysis of a potentially much simpler discontinuum problem, with the problem being posed entirely in terms of the individual discontinuities which interconnect nodes laid out across the body under consideration. Additionally, when general purpose finite element programs are used to analyse the collapse state often relatively complex non-linear solvers are required, in contrast to the simpler linear programming
solvers generally required in the case of DLO.
Compared with non-linear FEM, DLO has the following advantages and disadvantages:
Advantages
Disadvantages
Engineering analysis
Engineering analysis involves the application of scientific analytic principles and processes to reveal the properties and state of the system, device or mechanism under study...
procedure which can be used to directly establish the amount of load
Structural load
Structural loads or actions are forces, deformations or accelerations applied to a structure or its components.Loads cause stresses, deformations and displacements in structures. Assessment of their effects is carried out by the methods of structural analysis...
that can be carried by a solid or structure prior to collapse. Using DLO the layout of failure planes, or 'discontinuities', in a collapsing solid or structure are identified using mathematical optimization methods (hence the name, 'discontinuity layout optimization'). It is assumed that failure occurs in a ductile or 'plastic
Plasticity (physics)
In physics and materials science, plasticity describes the deformation of a material undergoing non-reversible changes of shape in response to applied forces. For example, a solid piece of metal being bent or pounded into a new shape displays plasticity as permanent changes occur within the...
' manner.
How it works
The DLO procedure involves a number of steps, as outlined below.The set of potential discontinuities can include discontinuities which crossover one another, allowing complex failure patterns to be identified (e.g. involving ‘fan’ mechanisms, where many discontinuities radiate from a point).
DLO can be formulated in terms of equilibrium relations ('static' formulation) or in terms of displacements ('kinematic' formulation). In the latter case the objective of the mathematical optimization problem is to minimize the internal energy dissipated along discontinuities, subject to nodal compatibility constraints. This can be solved using efficient linear programming
Linear programming
Linear programming is a mathematical method for determining a way to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships...
techniques and, when combined with an algorithm originally developed for truss layout optimization problems, it has been found that modern computer power can be used to directly search through very large numbers of different failure mechanism topologies (up to approx. 21,000,000,000 different topologies on current generation PCs). A full description of the application of DLO to plane strain problems has been provided by Smith and Gilbert
DLO vs FEM
Whereas with finite element analysis (FEM), a widely used alternative engineering analysisEngineering analysis
Engineering analysis involves the application of scientific analytic principles and processes to reveal the properties and state of the system, device or mechanism under study...
procedure, mathematical relations are formed for the underlying continuum mechanics problem, DLO involves analysis of a potentially much simpler discontinuum problem, with the problem being posed entirely in terms of the individual discontinuities which interconnect nodes laid out across the body under consideration. Additionally, when general purpose finite element programs are used to analyse the collapse state often relatively complex non-linear solvers are required, in contrast to the simpler linear programming
Linear programming
Linear programming is a mathematical method for determining a way to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships...
solvers generally required in the case of DLO.
Compared with non-linear FEM, DLO has the following advantages and disadvantages:
Advantages
- The collapse state is analysed directly, without the need to iterate.
- Problems involving singularities in the stress or displacement fields can be dealt with without difficulty.
Disadvantages
- As with other limit analysis techniques, DLO provides no information about displacements (or stresses) prior to collapse.
- DLO is a relatively new technique so only a limited range of software tools are currently available.
Applications
DLO is perhaps most usefully applied to engineering problems where traditional hand calculations are difficult, or simplify the problem too much, but where recourse to more complex non-linear FEM is not justified. Applications include:- Analysis of geotechnical engineering problems (e.g. slope stabilitySlope stabilityThe field of slope stability encompasses the analysis of static and dynamic stability of slopes of earth and rock-fill dams, slopes of other types of embankments, excavated slopes, and natural slopes in soil and soft rock...
, bearing capacityBearing capacityIn geotechnical engineering, bearing capacity is the capacity of soil to support the loads applied to the ground. The bearing capacity of soil is the maximum average contact pressure between the foundation and the soil which should not produce shear failure in the soil...
or retaining wallRetaining wallRetaining walls are built in order to hold back earth which would otherwise move downwards. Their purpose is to stabilize slopes and provide useful areas at different elevations, e.g...
problems). - Analysis of concrete slabConcrete slabA concrete slab is a common structural element of modern buildings. Horizontal slabs of steel reinforced concrete, typically between 10 and 50 centimeters thick, are most often used to construct floors and ceilings, while thinner slabs are also used for exterior paving.In many domestic and...
problems. - Analysis of metal forming or extrusionExtrusionExtrusion is a process used to create objects of a fixed cross-sectional profile. A material is pushed or drawn through a die of the desired cross-section...
problems.
Software using Discontinuity Layout Optimization
- DLO MATLAB script (2009-) Provided by the CMD research group at the University of Sheffield, UK.
- LimitState:GEO (2008-) General purpose geotechnical software application.
External references
- DLO teaching resources provided by the Geotechnical Engineering Research Group at the University of Sheffield, UK.