Dynamic modelling - Revision history

Jpye at 16:09, 24 June 2012

2012-06-24T16:09:12Z

← Older revision		Revision as of 16:09, 24 June 2012
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	:<math>\mathbf{F}\left(\mathbf{y},\mathbf{\dot{y}},t,\mathbf{z}\right)=\mathbf{0}</math>		:<math>\mathbf{F}\left(\mathbf{y},\mathbf{\dot{y}},t,\mathbf{z}\right)=\mathbf{0}</math>

	where <math>\mathbf{z~~}</math> <math>\mathbf{,t~~}</math> are apparently non-differential variables (variables that appear in the equations without a corresponding derivative). Considering all variables <math>\mathbf{y}</math>, <math>\mathbf{\dot{y}}</math>, <math>t</math>, <math>\mathbf{z}</math> as algebraic, then the matrix <math>\mathbf{J}</math> can be		where <math>\mathbf{z}</math> are apparently non-differential variables (variables that appear in the equations without a corresponding derivative), and <math>t</math> is the (non-differential) independent (time) variable. Considering all variables <math>\mathbf{y}</math>, <math>\mathbf{\dot{y}}</math>, <math>t</math>, <math>\mathbf{z}</math> as algebraic, then the matrix <math>\mathbf{J}</math> can be
	calculated analytically (within some tolerance) if the following is true:		calculated analytically (within some tolerance) if the following is true:

SreenathaB: /* About ODE/DAE solvers */

2012-06-24T14:06:56Z

About ODE/DAE solvers

← Older revision		Revision as of 14:06, 24 June 2012
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	:<math>\mathbf{F}\left(\mathbf{y},\mathbf{\dot{y}},t,\mathbf{z}\right)=\mathbf{0}</math>		:<math>\mathbf{F}\left(\mathbf{y},\mathbf{\dot{y}},t,\mathbf{z}\right)=\mathbf{0}</math>

	where <math>\mathbf{z}</math>, <math>\mathbf{t}</math> are apparently non-differential variables (variables that appear in the equations without a corresponding derivative). Considering all variables <math>\mathbf{y}</math>, <math>\mathbf{\dot{y}}</math>, <math>t</math>, <math>\mathbf{z}</math> as algebraic, then the matrix <math>\mathbf{J}</math> can be		where <math>\mathbf{z}</math> <math>\mathbf{,t}</math> are apparently non-differential variables (variables that appear in the equations without a corresponding derivative). Considering all variables <math>\mathbf{y}</math>, <math>\mathbf{\dot{y}}</math>, <math>t</math>, <math>\mathbf{z}</math> as algebraic, then the matrix <math>\mathbf{J}</math> can be
	calculated analytically (within some tolerance) if the following is true:		calculated analytically (within some tolerance) if the following is true:

SreenathaB: /* About ODE/DAE solvers */

2012-06-24T14:05:48Z

About ODE/DAE solvers

← Older revision		Revision as of 14:05, 24 June 2012
Line 14:		Line 14:
	:<math>\mathbf{F}\left(\mathbf{y},\mathbf{\dot{y}},t,\mathbf{z}\right)=\mathbf{0}</math>		:<math>\mathbf{F}\left(\mathbf{y},\mathbf{\dot{y}},t,\mathbf{z}\right)=\mathbf{0}</math>

	where <math>\mathbf{z}</math> are apparently non-differential variables (variables that appear in the equations without a corresponding derivative). Considering all variables <math>\mathbf{y}</math>, <math>\mathbf{\dot{y}}</math>, <math>t</math>, <math>\mathbf{z}</math> as algebraic, then the matrix <math>\mathbf{J}</math> can be		where <math>\mathbf{z}</math>, <math>\mathbf{t}</math> are apparently non-differential variables (variables that appear in the equations without a corresponding derivative). Considering all variables <math>\mathbf{y}</math>, <math>\mathbf{\dot{y}}</math>, <math>t</math>, <math>\mathbf{z}</math> as algebraic, then the matrix <math>\mathbf{J}</math> can be
	calculated analytically (within some tolerance) if the following is true:		calculated analytically (within some tolerance) if the following is true:

Jpye: /* New syntax */

2012-02-16T02:00:34Z

New syntax

← Older revision		Revision as of 02:00, 16 February 2012
Line 62:		Line 62:

	== New syntax ==		== New syntax ==

			''For a full discussion on this topic, see [[Improved DAE syntax]]''

	[[Dante Stroe]] recently added support for [[LINK semantics]] that allows this fairly painful syntax to be improved, however the new code is not yet incorporated into the ASCEND 'trunk' (see [[VersionManagement\|svn info]]). Based on the LINK structure the following syntax can now be used to define derivative chains, for example:		[[Dante Stroe]] recently added support for [[LINK semantics]] that allows this fairly painful syntax to be improved, however the new code is not yet incorporated into the ASCEND 'trunk' (see [[VersionManagement\|svn info]]). Based on the LINK structure the following syntax can now be used to define derivative chains, for example:

BenAllan: /* New syntax */

2012-02-16T00:42:50Z

New syntax

← Older revision		Revision as of 00:42, 16 February 2012
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	* {{srcbranch\|dante\|models/johnpye/pendulum_wLINK.a4c}}		* {{srcbranch\|dante\|models/johnpye/pendulum_wLINK.a4c}}
	* {{srcbranch\|dante\|models/test/dopri5/dopri5test_wLINK.a4c}}		* {{srcbranch\|dante\|models/test/dopri5/dopri5test_wLINK.a4c}}

			An inherent weakness of the syntax separating INDEPENDENT from der, as given above, is that in a hierarchically or peerwise composed model, multiple definitions of the same independent variable are possible and may conflict (or this may be by modeler choice, with multiple integrals within an overall model being computed by an IVP solver in different phases of the same simulation process). Several workarounds are possible. All rely on there being a community, or at least a library, standard way of doing things at the modeling level and an IVP solver understanding the convention.
			* Syntactically: change the LINK-equivalent usage to be created from DER(dxdt,x,t) rather than DER(dxdt,x(solver search for independent and insert here)). Verbose but unambiguous. Solver pre-analysis still must verify that there is only one atom instance (t) that is the target of all DER definitions, even if that instance has multiple names and declarations that may have been merged.
			* Typing an ATOM type to enforce that there exactly 1 independent: UNIVERSAL ATOM time_domain REFINES time; All instances of time_domain will automatically be the same. IVP Solvers can examine atom types and identify independent variables as those of type *_domain, eliminating the INDEPENDENT declaration. This solution actually allows multiple domains, and a parameter to an IVP analysis routines is "which domain type" is the independent variable. This would allow switching integrals over space(1-d) and integrals over time in the same model, an interesting capability in Low-Order PDEs. It might also allow interesting analyses in Fourier spaces.

			Current DER style doesn't scale to 2 simultaneous independent dimensions: INDEPENDENT x,y; DER(pdUxy,U). In a simultaneous 2-d world, the universal atom approach fails. One can (and probably JP has) argue that PDE models must explicitly represent the grid at the ascend language modeling level, as in the BVP library. We note, however, that NO current production PDE-oriented DSL actually works this way: they all make declarations based on sets and boundaries with arbitrary external definition whose consistency is checked by the solver.


	See also		See also

Jpye: /* About ODE/DAE solvers */

2011-03-28T23:13:09Z

About ODE/DAE solvers

← Older revision		Revision as of 23:13, 28 March 2011
Line 14:		Line 14:
	:<math>\mathbf{F}\left(\mathbf{y},\mathbf{\dot{y}},t,\mathbf{z}\right)=\mathbf{0}</math>		:<math>\mathbf{F}\left(\mathbf{y},\mathbf{\dot{y}},t,\mathbf{z}\right)=\mathbf{0}</math>

	where <math>z</math> are apparently non-differential variables (variables that appear in the equations without a corresponding derivative). Considering all variables <math>\mathbf{y}</math>, <math>\mathbf{\dot{y}}</math>, <math>t</math>, <math>\mathbf{z}</math> as algebraic, then the matrix <math>\mathbf{J}</math> can be		where <math>\mathbf{z}</math> are apparently non-differential variables (variables that appear in the equations without a corresponding derivative). Considering all variables <math>\mathbf{y}</math>, <math>\mathbf{\dot{y}}</math>, <math>t</math>, <math>\mathbf{z}</math> as algebraic, then the matrix <math>\mathbf{J}</math> can be
	calculated analytically (within some tolerance) if the following is true:		calculated analytically (within some tolerance) if the following is true:

Jpye: /* About ODE/DAE solvers */

2011-03-28T23:12:53Z

About ODE/DAE solvers

← Older revision		Revision as of 23:12, 28 March 2011
Line 14:		Line 14:
	:<math>\mathbf{F}\left(\mathbf{y},\mathbf{\dot{y}},t,\mathbf{z}\right)=\mathbf{0}</math>		:<math>\mathbf{F}\left(\mathbf{y},\mathbf{\dot{y}},t,\mathbf{z}\right)=\mathbf{0}</math>

	where <math>z</math> are apparently non-differential variables (variables that appear in the equations without a corresponding derivative). Considering all variables <math>\mathbf{y}</math>, <math>\mathbf{\dot{y}}</math>, t, <math>\mathbf{z}</math> as algebraic, then the matrix <math>\mathbf{J}</math> can be		where <math>z</math> are apparently non-differential variables (variables that appear in the equations without a corresponding derivative). Considering all variables <math>\mathbf{y}</math>, <math>\mathbf{\dot{y}}</math>, <math>t</math>, <math>\mathbf{z}</math> as algebraic, then the matrix <math>\mathbf{J}</math> can be
	calculated analytically (within some tolerance) if the following is true:		calculated analytically (within some tolerance) if the following is true:

Jpye: /* About ODE/DAE solvers */

2011-03-28T23:11:17Z

About ODE/DAE solvers

← Older revision		Revision as of 23:11, 28 March 2011
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	With the above technique, it may turn out that <math>\mathbf{J}</math> is singular, even under [[SVD]]. This is the usual first indicator of a numerical index problem.		With the above technique, it may turn out that <math>\mathbf{J}</math> is singular, even under [[SVD]]. This is the usual first indicator of a numerical index problem.

	It turns out that in many cases the index problem is even visible structurally in [[QRSlv]] -- meaning there is no feasible way to form <math>\mathbf{F}\left(\mathbf{\dot{y}, \mathbf{z}_\mathrm{free}\right)</math> by computing all the variables <math>\mathrm{\dot{y}}</math> expected and fixing all the variables <math>\mathbf{y}</math> expected. In this case either the problem is badly posed or a proper ODE problem can be recovered by moving some pair <math>\dot{y}_i, y_i</math> into the set <math>\mathbf{z}</math> and perhaps adding extra constraints obtainable by differentiation of a subset of <math>\mathbf{F}</math>.		It turns out that in many cases the index problem is even visible structurally in [[QRSlv]] -- meaning there is no feasible way to form <math>\mathbf{F}\left(\mathbf{\dot{y}}, \mathbf{z}_\mathrm{free}\right)</math> by computing all the variables <math>\mathrm{\dot{y}}</math> expected and fixing all the variables <math>\mathbf{y}</math> expected. In this case either the problem is badly posed or a proper ODE problem can be recovered by moving some pair <math>\dot{y}_i, y_i</math> into the set <math>\mathbf{z}</math> and perhaps adding extra constraints obtainable by differentiation of a subset of <math>\mathbf{F}</math>.

	DASSL and IDA work around low index problems by solving all model and stepping simultaneously, but for problems typical in chemical engineering (setting pressure		DASSL and IDA work around low index problems by solving all model and stepping simultaneously, but for problems typical in chemical engineering (setting pressure

Jpye: /* About ODE/DAE solvers */

2011-03-28T23:10:57Z

About ODE/DAE solvers

← Older revision		Revision as of 23:10, 28 March 2011
Line 28:		Line 28:
	With the above technique, it may turn out that <math>\mathbf{J}</math> is singular, even under [[SVD]]. This is the usual first indicator of a numerical index problem.		With the above technique, it may turn out that <math>\mathbf{J}</math> is singular, even under [[SVD]]. This is the usual first indicator of a numerical index problem.

	It turns out that in many cases the index problem is even visible structurally in [[QRSlv]] -- meaning there is no feasible way to form <math>\mathbf{F}\left(\mathbf{\dot{y}, \mathbf{z}_\~~mathrf~~{free}\right)</math> by computing all the variables <math>\mathrm{\dot{y}}</math> expected and fixing all the variables <math>\mathbf{y}</math> expected. In this case either the problem is badly posed or a proper ODE problem can be recovered by moving some pair <math>\dot{y}_i, y_i</math> into the set <math>\mathbf{z}</math> and perhaps adding extra constraints obtainable by differentiation of a subset of <math>\mathbf{F}</math>.		It turns out that in many cases the index problem is even visible structurally in [[QRSlv]] -- meaning there is no feasible way to form <math>\mathbf{F}\left(\mathbf{\dot{y}, \mathbf{z}_\mathrm{free}\right)</math> by computing all the variables <math>\mathrm{\dot{y}}</math> expected and fixing all the variables <math>\mathbf{y}</math> expected. In this case either the problem is badly posed or a proper ODE problem can be recovered by moving some pair <math>\dot{y}_i, y_i</math> into the set <math>\mathbf{z}</math> and perhaps adding extra constraints obtainable by differentiation of a subset of <math>\mathbf{F}</math>.

	DASSL and IDA work around low index problems by solving all model and stepping simultaneously, but for problems typical in chemical engineering (setting pressure		DASSL and IDA work around low index problems by solving all model and stepping simultaneously, but for problems typical in chemical engineering (setting pressure

Jpye: /* About ODE/DAE solvers */

2011-03-28T23:09:14Z

About ODE/DAE solvers

← Older revision		Revision as of 23:09, 28 March 2011
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	This form is precisely the matrix needed at the initial condition, normally. So for each "function evaluation" of [[LSODE]], we solve the nonlinear problem		This form is precisely the matrix needed at the initial condition, normally. So for each "function evaluation" of [[LSODE]], we solve the nonlinear problem

	:<math>\mathbf{F} \left( \mathbf{\dot{y}}, \mathbf{z}_\mathrm{free}\right)=\mathbf{0}</math> such that st. <math>\mathbf{y}</math>, <math>t</math>, <math>\mathbf{y}</math>, <math>\mathbf{z}_\~~mathrf~~{fixed}</math> are all fixed		:<math>\mathbf{F} \left( \mathbf{\dot{y}}, \mathbf{z}_\mathrm{free}\right)=\mathbf{0}</math> such that st. <math>\mathbf{y}</math>, <math>t</math>, <math>\mathbf{y}</math>, <math>\mathbf{z}_\mathrm{fixed}</math> are all fixed

	and for each <math>\mathbf{J}</math> needed we take the Jacobian of <math>\mathbf{F}</math> and solve to get a column of sensitivity data. (In		and for each <math>\mathbf{J}</math> needed we take the Jacobian of <math>\mathbf{F}</math> and solve to get a column of sensitivity data. (In